The structure of strongly prime quadratic Jordan algebras

ADVANCES

69, 133-222 (1988)

IN MATHEMATICS

The Structure of Prime Quadratic Jordan

Strongly

Algebras

KEVIN MCCRIMMON Department University Charlottesville,

of Mathematics, of Virginia, Virginia 22903

AND EPHIM ZEL’MANOV Institute of Mathematics, Siberian Division, Academy of Sciences of the USSR, Novosibirsk 90, USSR

We extend the structure theory for strongly prime Jordan algebras of arbitrary dimension to quadratic algebras of characteristic 2: we show that such J are either (1) hermitian forms H,(A, *) aJcH(Q(A), *) lying between an ample subspace of hermitian elements in a *-prime associative algebra A and those in its Martindale ring of quotients Q(A), (2) Clifford forms with a scalar extension ajz J which is an algebra &=J(Q, 1) of a nondegenerate quadratic form Q with basepoint 1, or (3) albert forms with a scalar extension aj,J which is a 27-dimensional split albert algebra Llj= H,(K(SZ)) of 3 x 3 hermitian matrices over an 8-dimensional split octonion algebra K(0). As a consequence, the simple Jordan algebras (of arbitrary dimension) are either hermitian H,(A, *) for *-simple A, or Clifford forms in J(Q, 1), or are albert algebras. In particular, our methods give an idempotent-free classification of all simple finite-dimensional algebras which are i-special. The key to this characterization is the existence of Zel’manov polynomials in the free special Jordan algebra, which are both hermitian polynomials (whose values always look like hermitian elements) and at the same time Clifford polynomials (nontrivial on 3 x 3 hermitian matrices, hence on special algebras with 23 interconnected idempotents): if a Zel’manov polynomial does not vanish on J then J is of hermitian type, and if it does vanish on J then J is of Clifford type. One example of such a polynomial is 448 = [ b16(x1, yl, Zlr wd~P16(x2~Y2~ z21 w2)l~P~6(x3~Y3~z3? w3)1 for P16=

CCD:,(Z)~, D,.,(w)l. D,.,(w)1 (D,,(z)= CCX,YI,~1).Thusthe structureof

Jordan algebras depends very directly on polynomial identities and commutators. 0 1988 Academic

Press, Inc.

Contents. Introduction. 0. Review. Part I. Algebras of hermitian type. 1. Special prime algebras with hermitian part. 2. Hermitian absorption. 3. Splitting semiprime algebras into anti-hermitians and special hermitians. Part II. Algebras of Clifford type. 4. Central closures. 5. Scalars in Clifford algebras. 6. Clifford forms. 7. Clifford identities. 8. Centrality of [x. y12. 9. Algebras with [x, y12 ~0. 10. Algebras with

133 OOOl-8708/88 $7.50 Copyright 0 1988 by Academic Press, Inc. All rights of reproduction in any form reserved.

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[x. y]* E @l. 11. Prime algebras of Clifford type. Part III. Construction of Zefmanou 12. Eaters. 13. Hearty eaters. 14. Construction of hearty eaters. Purr IV. theorems. 15. Classification theorems. Part V. Radicals. 16. Radical identities. 17. Radicals of P.I. algebras. Index of definitions. Index of notations.

polynomials. Classification

INTRODUCTION

The classical structure theory of linear Jordan algebras with finiteness conditions was complete by 1965 [3], and the extension to quadratic algebras by 1969 [4]; the analysis into the three simple types (hermitian, Clifford, albert) depended on properties of Peirce decompositions relative to idempotents, and hence required some sort of finiteness. In hindsight it is now clear that the proper finiteness condition is not finite-dimensionality or d.c.c. on inner ideals, but rather the property of having a capacity. In 1979 [29] the second author showed that prime nil-free linear Jordan algebras (without any finiteness hypotheses whatsoever) either were albert forms or were i-special. In 1983 [32] he showed that “nil-free” could be reduced to “nondegenerate” (which is the natural hypothesis in view of some degenerate examples of Pchelintsev [24]), and that the i-special ones were necessarily of either hermitian or Clifford type. In particular, this gave a very precise description of all simple or division algebras of arbitrary dimension. (The second author had already given a description of division algebras in 1979 [30] by a more delicate and less general argument.) In 1982 [9] the first author carried over the splitting into albert and i-special algebras to quadratic algebras. In the present work we carry over the splitting of i-special algebras into hermitian and Clifford types to quadratic algebras over arbitrary rings of scalars. This gives us a clear picture of prime nondegenerate Jordan algebras in general. In order to stress how completely the theory depends on the mere existence of certain polynomial identities or non-identities, and not on their paticular form, we immediately obtain the hermitian structure theory in Part I in a few pages. An ideal S(X) 4 d(X) in the free special Jordan algebra on an infinite set of indeterminates is “hermitian” if it is closed under all n-tads {x1 . . . x, } = x1 I . . X, + x, . . . xi and is therefore a “universally hermitian” algebra (the values X’(J) taken on by these polynomials on any special J form an ideal of hermitian elements, S(J) z H,(A, *). In Section 1 we show that this fact together with some generalities about associative algebras establishes the structure of prime special algebras having nonzero hermitian part which are nondegenerate (or even “hereditarily semiprime”): they have a fat hermitian part H,,(A, *) = Z(J) 4 J, and by absorption all of J fits inside the Martindale ring of quotients Q(A) of A, so J c H( Q(A), *) where both A and Q(A) are *-prime. Note that if J is

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135

simple then this fat ideal must be all of .Z, so J= Z-Z,,(A,*) is a full hermitian algebra. In Section 2 we show that such an 2 is “absorptive”: if la Jc H(A, *) then any element in .ZnYA(Z) generated associatively out of Z will be absorbed back into Z by a high power of 2. Now the condition .Zn YA(Z) c Z is what is needed to guarantee that J/Z remains special, and we use this absorptive property to show in Section 3 that any semiprime i-special algebra splits as a subdirect sum of a special algebra in which Z(J) is “dense” and an algebra where all 2 = 0; in particular, prime i-special algebras are either of anti-hermitian type (X E 0) or special hermitian type (X(J) dense). The Hermitian Structure Theorem asserts that the prime algebras of hermitian type which are i-special and hereditarily semiprime are necessarily hermitian forms. Thus prime i-special algebras must have hermitian form as soon as they admit nonzero values of hermitian polynomials such as qa8. In anti-hermitian algebras qd8 = 0 vanishes identically, and in Part II we make a general study of algebras of Clifford type (satisfying a “Clifford identity” f= 0, one not satisfied by hermitian 3 x 3 matrices). We begin in Section 4 by showing that central closures can be constructed (but with surprising difficulty in characteristic 2). In Sections 5 and 6 we observe some properties of Clifford algebras and describe their forms. In Section 7 we show that a nondegenerate i-special algebra which satisfies a Clifford identity satisfies the “standard” Clifford identity [[x, y]‘, z, w] = 0 (even [x, y] * 0z = 0 in characteristic 2) and therefore a doubly interconnected algebra cannot satisfy any Clifford identity whatsoever. In Section 8 we show that the standard Clifford identity forces [x, y]* to be central (this is trivial for linear Jordan algebras, but arduous for quadratic algebras). In Section 9 we show that nondegenerate algebras with [x, y]* G 0 are “degree 1,” JC sZ+ for a commutative associative Q. In Section 10 we show that algebras with [x, y]* f 0 scalars in @ are Clifford algebras J(Q, 1) of quadratic forms over @ (this follows immediately for linear algebras from the Hall-Zelmanov Identity 2[x, y]* 0 x2 - ( [x, x or] o [x, y] ) OX + (2[x, xoy]* - [x, y] 0 [x, x2 oy]) 1 = 0, but for quadratic algebras we get only a degree 4 polynomial in x, and must assume @ is algebraically closed to argue this down to degree 2). The Clifford Structure Theorem in Section 11 asserts that nondegenerate i-special prime algebras of Clifford type are Clifford forms: they become Clifford algebras J(Q, 1) under scalar extension. In particular, they too are special. Only in Part III do we actually exhibit a Zel’manov polynomial qd8 which is both hermitian and Clifford. In Section 12 we show that for linear algebras one hermitian ideal is the biggest ideal T4(X) which “eats tetrads” (when we put it into a tetrad it dissolves the tetrad into Jordan products), consisting precisely of all pentad-eating elements. In the quadratic case we use “hearty tetrad eaters” (those that eat everything resembling tetrads); in

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Section 13 we show the hearty tetrad-eating ideal is only slightly larger than the hearty pentad-eating ideal H,(X), where the latter consists precisely of all hearty pentad-eating elements. In Section 14 we show quite generally that commutators of hearty tetrad-eaters are hearty pentadeaters, and we construct hearty tetrad-eaters (such as P,~) by repeated commutation with [x, y]. For Martindale’s theorem on the extension of Jordan homomorphisms, it is important that as soon as an i-special algebra is doubly interconnected it is entirely Zel’manovian, Qd8(J) = J for Q&X) c H,(X) the hermitian ideal generated by qd8. In Part IV (Section 15) we harvest the structural consequences, obtaining the main structure theorems for prime, simple, and division Jordan algebras: they are of hermitian, Clifford, or albert type. In Part V we draw some conclusions about radicals. In Section 16 we show that, as in alternative algebras, the free Jordan algebra has a radical consisting of the elements which refuse to assume nonzero values in any respectable Jordan algebra (they vanish on all special and all albert algebras), and in consequence are condemned to take on radical values wherever they go (their values on any Jordan algebra J lie in the radical of J). Originally [32] this radical result was needed to extend the dichotomy between i-special and albert algebras from nil-free to nondegenerate algebras (a somewhat unsatisfactory situation, since the radical result in turn needed the dichotomy for nil-free algebras), but this extension can now be done more directly [8]. In Section 17 we extend to quadratic algebras the result [ 311 that the strictly nil and degenerate radicals coincide for any Jordan P.I. algebra. Because of the length of this paper, we include for the reader’s convenience indexes of definitions and notations. 0. REVIEW

We remind the reader of the basic quadratic notations; as references we mention the books [2,4,7]. A unital (quadratic) Jordan algebra J= (J, U, 1) over an arbitrary ring CDof scalars consists of a @-module J, a distinguished element 1 E J, and a quadratic map U: J+ End,(J) such that if we denote the linearization of U by Vx,Jz)=

ixJJz>=

U,,,(Y)

(U,,.=

ux,,-

then (0.1) (0.2)

(0.3)

U1=Id ux vy, .x= vx, y ux = U”(x,y, x U U(x)y = ux u, ux

ux-

Uz)

STRONGLY

hold strictly extensions (0.4)

PRIME

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JORDAN

in the sense that they continue

Jn= J&Q:

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ALGEBRAS

to hold in all (free) scalar

uxx,,,, = 1 U, 0 0: + C Uxt,x,0 I i
Wi"j.

(Equivalently, all linearizations of these identities hold in J itself.) We will often consider the polynomial extension (0.4’)

J[T]=J@@@[T]

consisting of all polynomials in the scalar indeterminates t E T with coefficients from J. A (quadratic) Jordan algebra is just a subspace J= (J, U, 2) of some unital Jordan algebra closed under the products U,y and the square x2 = U,l,

(0.5)

in which case J imbeds in the particular free unital hull (0*6).?=@1@J:

u al~x(p10Y)=~2PO{~2Y+2~px+~x~y+PX2+U.~Y},

where we denote the linearization Vx(Y)=xoY=

K,,

of the square by 1

(= (x+y)2-x2-y2).

If $e @ we can characterize these algebras axiomatically Jordan algebras with product x . y = 4 x 0y satisfying x.y=y.x,

as the linear

(x2.y).x=x2.(y.x).

The most important Jordan algebras come from associative algebras. Any associative algebra A yields a quadrative Jordan algebra via (0.7)

A + : U, y = xyx,

x2 = xx,

{ xyz } = xyz + zyx,

xc’y=xy+yx

(which is unital if A is). A Jordan algebra is special if it is isomorphic to a Jordan subalgebra of some A +. An important example is a hermitian algebra (0.8)

H(A,*)=(x~Alx*=x}cA+

of self-adjoint elements in an associative algebra with involution example, the algebra A = M,(D)

*. For

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of n x n matrices with entries in D has conjugate transpose involution X= (d,) + X* = (djj) if d-+ d is an involution on D. More generally, we must consider ample hermitian algebras (0.8’)

H,(A, *) c H(A, *): aH,a* c H, for all a E A, and all traces and norms aa* lie in H, (or equivalently, if A is unital, 1 E H,).

a+a*

If 4 E @ then the only ample subspace is Ho = H, but in general there are many different ones (e.g., in M,(Q) for Q quaternion of characteristic 2 we have H,, consisting of all hermitian matrices with scalar diagonal entries). Another important algebra is a (Jordan) cZifSord algebra, which lives in the associative Clifford algebra C( Q, 1) of a quadratic form 52 with basepoint 1: (0.9)

J(Q, l)cC(Q,

U,Y=Q~,J,X-Q(X>.T

1):

(r=Qb

1) 1-y).

An associative algebra A is an (associatioe) envelope for a special Jordan algebra Jc A if it is generated as an associative algebra by the elements of J, and an associative algebra with involution (A, *) is a *-envelope for J if it is generated by Jc H(A, *). An algebra is i-special if it satisfies ail the Jordan identities of special algebras (equivalently, is a homomorphic image of a special algebra). An algebra is exceptional if it is not special, and i-exceptional if it is not even i-special. The basic i-exceptional algebras are the 27-dimensional albert algebras (0.10)

J(N, 1):

(Thy)=

U,y=T(x,y)x-x”

-&$,hgNI,,

xy

T(x”A=~,N,)

determined by certain nondegenerate “admissible” cubic forms N. The most important froms N are given by the Tits Constructions. The First Tits Construction [Ill, Theorem 6, p. 5073 (0.101)

JV, P):

J=A@A@A,

1 = (40, Oh

N(a,,al,a,)=n(a,)+~nn(al)+~--‘n(a,)-t(a,ala,)

starts from a central simple degree 3 algebra A over a field @, with generic norm n and trace t, and a scalar 0 #p E @‘; the Second Tits Construction [ 11, Theorem 7, p. 5091 (0.1011)

JM *, u, PL):

J=H(A,

*)@A,

Nao, a) = n(aO) + ,un(a) + p*n(a*)

1 = (1, Oh - t(a,aua*)

starts from a central simple degree 3 algebra A over a field Sz with

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involution * of second kind, with Q separable quadratic over @= H(R, *), and U=U* an element with n(u)=pp* for some O#~(E&?. We have the usual categorical paraphernalia. A homomorphism T: J + J’ of Jordan algebras preserves U-products and squares (or, in the category of unital algebras, the unit), (0.11)

T(&Y)=

(or T(l)=

T(x’) = TV

U,,,W),

1).

A derivation D on J has (0.12)

D(U,y)=

UDc,,,~+

U.Jb),

D(x2)=D(xbx

(orD(l)=O).

We have direct products n Ji and subdirect products J+j

(Jj = xi(J) for homomorphisms

J,

rci with n Ker xi = 0).

We denote the algebra direct sum J, x J, as

J,

q J2

(as opposed to a module direct sum J, 0 J2). The kernels of homomorphisms are precisely the ideals la spaces which are simultaneously (0.13)

J,

inner and outer ideals

B 4 Jinner:

U,Jc

B

(i.e., U,J+

B2 c B),

Jouter:

UjKc

K

(i.e., U,K+

V,Kc

Kd

K).

We can factor out ideals via the usual quotient algebras J/Z. Inner ideals are the analogues of one-sided ideals in the associative theory. If 1 E @ the outer ideals are the same as ideals, but if t# 0 we can have proper ample outer ideals

(0.13’)

JO + J

(outer ideals containing 1); for example, an ample JO = H,(A, *) in (0.8’) is an ample outer ideal in J= H(A, *). We use the notation

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to denote the Jordan ideal in J generated by the subset XC J, and by x4(X> the associative ideal in A generated by a subset XC JC A+. We denote the Jordan subalgebra of J generated by X (when J is understood) by

In the other direction, we call the biggest ideal of J contained in a subspace K the core of K, and denote it by p=c

{Illa

J,ZcK}.

The product of ideals is again an ideal Z,KaJ

and in particular (0.14)

=z.

U,KaJ,

we have a decreasing chain of derived ideals ztn) = D”(Z)

(D(K)

= K3 = U,K).

If Z is an ideal so are its two-sided and inner annihilators (0.15) (0.15’)

Ann,(Z)=(z~J~U,Z=U,z=(zZJ)=O)aJ Inann,(Z)={z~JJU,z=U,U,.?=O}aJ.

An algebra is semiprime if it has no self-orthogonal ideals U,Z= 0 (equivalently, no solvable ideals I’“’ = 0). This is further equivalent to being a subdirect sum of prime algebras, where an algebra is prime if it has no orthogonal ideals U,K = 0, equivalently [ 161 if there are no annihilators (0 # Z a J* Inann, = 0), or equivalently if the nonzero ideals have the finite intersection property (0 # Z, K a J* 0 # In K), is simple if it has no proper ideals at all (and is not trivial, J* = U, J = 0), is a division algebra if all its nonzero elements are invertible (x # 0 => U, is invertible), is nondegenerate if it has no trivial elements U, =O, is strongly prime (or semiprime) if it is prime (or semiprime) and nondegenerate, is nil-free if it has no nil ideals consisting entirely of nilpotent elements z” = 0 (where the powers of z are defined recursively by z1 = z, z* = z*, and z”+* = Uzzn), is strictly nil if all scalar extensions remain nil (equivalently the polynomial extension J[T] remains nil), and is semiprimitive (semisimple) if it has no q.i. ideals consisting entirely of elements z with 1 -z invertible in J. In most cases there is a unique smallest ideal K whose quotient J/K has the desired property (the Baer or semiprime radical L%(J) creates semiprimeness; the nondegenerate radical Y(J) creates nondegeneracy, the

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nil radical Nil(J) creates nil-freedom, the strictly-nil radical SNil(J) creates strict nil freedom, and the Jacobson radical Rad(J) creates semiprimitivity). The structure of “nice” algebras turns out to depend very delicately on the properties and inter-relationships of all these radicals. One of the useful tools in the theory is the notion of a tight cover of J:JzJis tight if (0.16)

all ideals of 7 hit J

(O#f-a~*0#7nJ).

The importance of this sort of cover is that it automatically theoretic structure, (0.17) if J is prime, semiprime, nondegenerate, nilfree, then so is any tight cover s.

inherits ideal-

nil-free, or strictly-

(for nondegeneracy see [ 19, (2.9)]). We have similar notions for associative envelopes of Jordan algebras. An associative +-envelope A is a *-tight envelope for J if (0.18)

all *-ideals of A hit J

(O#B*aAaO#BnJ).

Again we have (cf. [ 143 ) (0.19) if J is simple, prime, semiprime, or nil-free then any *-tight envelope A for J is *-simple, *-prime, *-semiprime, or nil-*-free. (Tight covers also interact well with the Jacobson radical, by unpublished work of the second author.) We will make frequent use of tight unital-scalar extensions 3=&j, obtained from a unital hull .? of J by extending the scalars from @ to 8 (but tightly, not freely as in (0.4)-5” is spanned over 3 by 1 and J). Some consequences of the axioms (0.1 k(O.3) which we will need later are

(0.20) U{x.v}+ UU(.x) U(y)2,z= uxu.vuz+ wJyux+ K,,U,V,,, (0.20’)

ux,z+

(0.21) (0.22)

v.~..~~+~,v,,,=~~,,),:, {xxz}=x*oz,

(0.23)

2u, = v’, -

U”(x)r,s=

vxu;+ uzv,= {xyz}+{yxz}=(xoy)oz rfx2,

V Wxb, ” = Vx, U(y)x3 (0.24)

u, uz + u; u, + v, u; v, u,.,.

vz.y=vzvy- uz., vx,.“vz,y=

vx,u(y)l+

K, y K. i = VU(.Qy,i + ux uy, Z’

ux,zuy9

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We are just beginning to see how important commutators are in Jordan algebras, even for linear Jordan algebras defined by a commutative product! Although there is no element [x, y] in a Jordan algebra, the ghosts of these elements hover and play a crucial role in the theory. The case of special Jordan algebras makes it reasonable to define (0.25)

[x, y12 := x 0 t&x - u, y2 - qx”

(0.26)

u [x. VI := w-x, y - u.x uy - uy u.x

( w.x, y = V.X..v vy. x - VU(x) 1’2= U.Y y - u.x uy - uy ux = ut., y - K*, y4 (0.27)

CC~,Yl, zl := {XYZ) - {YXZ> = CK, VJ z :=4,(z).

These behave as if they were live elements: here D,,, is the standard inner by x and y, and is given in terms of the 0 -associator by

derivation determined [a,b,c]“=(aob)oc-ao(boc) (0.27’)

D,,,(z) = - Cx, z, YI”,

and we have (0.28) (0.29) (0.30)

(0.31)

0:

.” = V,& Yl

* - 2u,,,,

UC&y, 1 = [XT Y12 cc~~Yl~~12=~~~~x,~,~-

~~.x,y,z2--u,c4Y12

uccx, y], 21= UU(Cx,.v])z,i - u,.x, y] uz - uz u,,

y]

-Dx,yuA.y.

Recently Thedy [28] s-identity (0.32)

has shown that commutators { U”,,,

y]z

- u,.x. y, uz u,,

y]

give rise to a new

1w = 0

true in all special algebras but not in albert algebras. This shows that commutators can be used to distinguish between special and exceptional algebras. That such elements are intimately related to speciality is clear from [x,y]~[z,w]-4{xyzw}=(xoy)o(z~w)-2{x~yzw} -2{xyzow},

where neither the product [x, y] 0 [z, w] nor the tetrad {xyzw} can by itself be expressed in Jordan terms.

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Rather than derive all these identities from the axioms, we can obtain most of them by appeal to Macdonald’s principle (0.33)

if an identityf(x,

y, z) = 0 of degree < 1 in z holds

in all algebras A + for A associative, then it holds in all Jordan algebras J. The axioms (0.30) and (0.31) are derived in [21, (3.1)]. The case of identities f(x, y) = 0 in two variables is Shirshou’sprinciple. PART I.

ALGEBRAS

OF HERMITIAN

TYPE

Hermitian polynomials are Jordan polynomials whose values are “intrinsically hermitian,” which in practical terms means that they dissolve tetrads into Jordan products. Until the works [30, 321 it would have been thought preposterous to believe in the existence of such creatures; the first intimation that tetrads were not as far removed from being Jordan as people thought was the identity [30] UccJ,, X212,rX,, X43,{x1 x2x3x4} = p(x,, x1, x3, x4) for a Jordan polynomial p, and in [32] the first hermitian polynomial t(x,, x2, x3, x4) (of degree 2160!) was exhibited: u r(x,,-r2,.q,x4) In this part we Lh Y2Jwd ‘P( x1,x2, x3,x4,yl,y2,y3~~d trace the elegant straightforward influence that hermitian polynomials have on a prime Jordan algebra: semiprime i-special algebras split into an antihermitian part where all hermitian polynomials vanish, and a hermitian part where the values of hermitian polynomials are dense. Prime algebras of hermitian type are actually algebras of hermitian elements. Everything flows naturally and almost immediately from the definitions. Later, in Part III, we justify faith in these miraculous creatures by exhibiting several.

1. SPECIAL PRIME ALGEBRAS WITH HERMITIAN

PART

Hermitian polynomials are those that belong to hermitian ideals, where an ideal &’ of polynomials in the free special Jordan algebra is hermitian if it is n-tad closed for all n. This definition guarantees that the values X’(J) assumed by GK?on any special Jordan algebra Jc H(A, *) have the form x(J) = &,(A,, *) of an ample space of hermitian elements in the subalgebra A0 generated by X(J). From this and basic facts about Martindale quotient rings, it is easy to show why a prime special algebra J with hermitian part must look hermitian, H,(A, *) 4 JC H(Q(A), *).

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Let X be an infinite set of indeterminates, f(X) = FSJ(X) the free special Jordan @-algebra on X. This has as its special universal envelope the free associative algebra a(X) on X; here f(X) c H(a(X), *) is the Jordan subalgebra generated by X, where the standard or main involution * on a(X) is uniquely determined by the condition that the generators be symmetric: x* =x for all x E X. An ideal 9 a f(X) is called formal if its elements depend on their form rather than the particular variables appearing in them, (1.1)

P(XI 9 ***,x,) E 9 =a P(dXl), *.a,&J) for all permutations 0 of X.

A formal ideal S(X)

a f(X)

(1.2)

{~~~~~}c~

E JJ

is hermitian if it is n-tad closed for all n 2 4, foralln>4,

wherethen-tads {a,...a,}=a,~~~a,+a, . . . a, are symmetric polynomials in H(a(X), *) which are not Jordan for n > 4. (For n < 3 note {ui} = 2a,, {al~2~3} = {u,u~u~}.) For example, %=O is trivially (4%) =Q,oQ,, hermitian and &‘= f(X) is not (since {xIx~x~x~} belongs to H(a(X), *) but not to y(X)). However, by Cohn’s theorem [3,26] when 4~ 0 all n-tads can be expressed as Jordan products of xis and tetrads {xi,xi~xi~xi.,} for i, < i, < i, < id, so in this case closure under tetrads forces closure under all n-tads: in linear Jordan algebras 2 is hermitian iff (1.21in)

(Jif~~2f}c2f

iffE@.

A polynomial p(x,, .... x,) E f(X) is called hermitian if it lies in some hermitian ideal X(X) (we do not demand that the ideal Y(p) generated by p itself be hermitian). For any i-special Jordan algebra J we can evaluate the polynomials in $ on J, and the values taken on by the polynomials in %(X) form an ideal X(J)

a J

called the &‘-part of J. (In general, when f a j(X) is formal and X is infinite the values 9(J) form an ideal in J: if t(x,, .... x,) and t’(x,, .... x,) lie in 9 then t(a,, .... a,) + t’(bI, .... b,) = {t(x,, .... x,) + f’(x,,+ ,, .... x,,+?)} (a 1, .... a,, b,, .... b,), ah = {K,+,fh *.., -a)(%

.... a,) = (at(x,,...,x,)}(a,, .... a,), Uat(a,,...,4 ***9a,, ah Utca,, .... ,)a = { uth. .... x.)xn+ I >(a 1, ....

a,, a), where we need X infinite to guarantee independent x,+ , , .... x, + r exist and we need formality (1.1) to guarantee that t’(x,, .... x,) E 9 * f(Xn+L, **a,X,+,)E~.) Note that such evaluation ideals 9(J) are always invariant under T of J as in (0.11): T(p(a,, .... a,)) = all algebra endomorphisms P( T(a, ), .... T(4)

E P(J).

STRONGLY

PRIME

QUADRATIC

The reason for the term “hermitian (1.3)

if Jc H(A, *) for hermitian ideal %(J)=H,(A,, *) the subalgebra of

This follows from

JORDAN

145

ALGEBRAS

ideal” is the fact

some associative *-algebra A, then for any H the X-part of J has the form of an ample subspace of H(&, *), for A, A generated by 2(J).

the quadratic

version of Cohn’s

theorem [13]: if *) is ample in H(Ao, *) as in (0.8’) for A0 the associative subalgebra of A generated by K. The n-tad closure of X(J) follows from n-tad closure of 2’(X): if {hi(X,)...h,(X,)} =h(X,, .... X,) (Xi= {Xii, .... x,(i)}) then {h,(a,,, .... %(I))> , , .,.,‘**T x hjE(%I,, I -.-9%(n) j h(l;J = h(q,:, .... a,,(,,) E z(J) (usfng again that ,+,)EX(X) by formality (1.1) (and the W fact that kch an identity’ifn’kk free associative algebra a(X) implies the same in any associative algebra A). This hermitianity forces these polynomials to vanish on Clifford algebras Kc H(A, *) is n-tad closed for all n > 4 then K= H,(A,,

(1.4)

any hermitian polynomial vanishes on a nondegenerate ford algebra J(Q, 1) of dimension 2 5

clif-

since if the ideal 2(J) is not zero it must be all of the simple algebra J= J(Q, l), yet J(Q, 1) # H,(A, *) for the envelope A = C(Q, 1) when dim J25. Before we establish the main result we will need a general fact about *-tight envelopes of ideals. 1.5. LEMMA. Let A” be a *-tight generated by an ideal I a 7. Assume

envelope for

Inann~(Z(m’) = 0

for

J, A the subalgebra

all m.

Then A is a *-tight envelope for I. Proof:

(1)

Suppose B is a *-ideal of A with BnZ=O.

To prove B = 0 it s&ices to prove B = 0 for B = 32(B) = uFzO $‘B$’ the *-ideal of 2 generated by B (recall that 7 generates the envelope A), and by *-tightness of 2 it suffices to prove BnT= 0. So consider any element Z-EBnJ, say (2)

2 E ?‘B?‘.

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MCCRIMMON AND ZEL'MANOV

By our hypothesis we will have 2 = 0 if we can show 2 E Inannl(Z’”

(*I

To see that I belongs to this annihilator

+ ‘)). (0.15’), it suffices by (1) to show

Now the left side of (**) clearly lies in Z (indeed, in Z(n+2) a J), so it suffkes to show it lies in B. For this we first observe a general absorption

(3) in terms of associative powers I” = II... for Ka J by .?(iJ,k’)=SLkk’k= {zkk’j u,z~K, so by induction zzz(l’1’2’...z(m~1)C~+l

Z (n factors), since &CC’) c KK k-k’(U,z)EKK by {zkk’},

~Z(m)=~~lJ(Z(m~l))(l)~~m-lJ(m~l)Z(m-l)~

7 and dually for (3). Using this, the left side of z'"+2'(~++~.)z(n+2)Cz(n+2)(~~~)z(n+2)C in Z(n+2)(jflB?n+2) ZCn+*) (by (2)) c 1”+3B1”+3 (by (3)) c ABA (since Z generates A) c B (since B a A), completing the proof of (**) and the lemma. 1

(**)

is

contained

The same holds if we assume Ann~(Z’“)) = 0 for all m, but the proof is more involved since there are three conditions (0.15) instead of two monomial conditions (0.15’). We say J is hereditarily semiprime if all its ideals remain semiprime. All nondegenerate algebras are hereditarily nondegenerate, hence hereditarily semiprime. In terms of hermitian algebras (0.8), (0.8’) we can state

1.6. SPECIAL HERMITIAN STRUCTURE THEOREM. A special algebra which is prime and hereditarily semiprime and has a nonzero hermitian part SF(J) #O is a hermitian form: it looks like an algebra of hermitian elements, &(A, 4 =x(J) for a *-prime associative Q(A).

a Jc WQW, 4

algebra A with Martindale

*-algebra

of quotients

Proof: Since J is special, it has an associative *-envelope which we may tighten to get a *-tight envelope A” as in (0.18) (2 = E/M for A4 a maximal

147

STRONGLY PRIMEQUADRATICJORDANALGEBRAS

*-ideal missing Jc E, so all nonzero *-ideals of 2 hit .Z.) By (0.19), 2 is necessarily *-prime. Let A be the subalgebra of A generated by z(J) c J. By the basic hermitianity property (1.3)

S(J) = H,M *I is ample in H(A, *). We wish to show that A itself is *-prime. We know that the ideal Z= S?(J) a J remains semiprime by our Hypothesis that J is hereditarily semiprime, and therefore by [ 161 that Z= S’(J) is itself prime. We also know that Inann,(Z’“‘) = 0 by (0.15’) since J is prime and I# 0 =S I’“’ # 0 (by semiprimeness of J). Thus A is a *-tight envelope for Z by Lemma 1.5. Once A is a *-tight envelope for the prime algebra Z, by (0.19) it inherits *-primeness. But than we know from the associative theory that H,(A, *)=X(J) a Jc fZ(Q(A), *): in general la Jc Q(A) for any nonzero ideal Z in a prime special J and any *-tight envelope A of Z (indeed, 2 c Q(A) for any *-tight envelope 2 for J), where Q(A) = QF,n(A) is the Martindale algebra of symmetric quotients relative to the filter of all nonzero *-ideals of A (see [20]). 1 Thus with surprisingly spreads its influence over Section 3 establishes the i-speciality to speciality polynomials.

little fuss, the existence of a hermitian part the entire special algebra. Our main result in same for an i-special algebra, but to reduce will require a deeper analysis of hermitian

2. HERMITIAN

ABSORPTION

The key fact for splitting semiprime i-special algebras into anti-hermitian and special hermitian parts is the absorptive nature of hermitian ideals: they absorb the associative ideal YA(Z) generated by Z back into the Jordan ideal I. It is also important that hermitian ideals eat n-tads; in Part III we construct a hermitian ideal by a careful analysis of eating habits. Besides intrinsic hermitianity, hermitian ideals are automatically absorptive. The T-absorber of an ideal K a J into an ideal Z a J for Jc 7 is (2.1)

AbsJ,J(K, I)=

{Sfl

U,K+

iJ,z+

{Kz?}

cZ}.

(This can also be viewed as the annihilator of K mod Z as in (0.15), but coming from J and not just J.) In terms of the derived ideals (0.14) of X(X) we have

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MCCRIMMON AND ZEL'MANOV

2.2. HERMITIAN ABSORPTION THEOREM. Zf&‘(X)a j(X) is a hermitiun ideal, then for any n a sufficiently high power of 2 eats n-tads, {y, . . . y,Yf(X)‘“‘}

(2.3)

for independent indeterminate (2.3’)

x(X)

Y) c $(Xu

. ..y.}csf(Xu

a d(X)

Y),

sets X, Y, and hence for all 0 < r 6 n

{Yl ~-YrJw-)‘“‘Y,+1

Whenever into f, (2.4)

c A?(Xu

satisfies

(2.3’)

Y)cy(xlJ it absorbs

Y).

into ideals, not just

if ZaJcA+ then {a,..~a,~(J)~“~a,+,~~~a,}cZn~(J) or a a, E j as long as at least one ai lies in Z, 1, ***, f

and it also absorbs envelopes: (2.5)

if Z 4 JC H = H(A, *), where A is generated by J, then for any a E H(A, *) n yA(Z) there exists n = n(u) such that

(i)

U,%(J)‘“‘+

{S(J)(“)aj}

cZn&(J)

andifaEt(9~(Z))={z+b+b*IzEZ,

(ii)

bEyA(Z)}

then

U,(,)(.)a c In x(J),

so

(iii) 2Hn yA(Z) + {.?j Hn yA(Z)} + UHnSACI) j c c LJ,“=o Abs,,,(X(J)‘“‘, In=%'(J)).

t(,a,(Z))

In particular

(iv)

{JI?JZ}

cAbs,,,(X(J)C2m+2),

In z(J)).

We have universal absorption (2.6)

*) then if My,, ...>y,) E H(a(Y), 1, . . . . Y,) E Abs H(a(xuY).*),~(x”Y)(~(~uY)(“), W

for

some

n = n(h), WXuY))*

Proof. Whenever HaJcA+ we have {...yH”‘...}c{...HH...} by (...yU,x’...}=(...{yxx’}x...}-(...x’U,y...} for U, YE H (cf. 1.5(3)), so by induction

{yxx’},

When H is n-tad closed this lies in {H H.. . H} c H. For H = ~(XU Y), J= f(Xu Y), A = b(Xu Y) we get (2.3). The property (2.3’) follows from (2.3)

by

induction

on

n,

using

{ ...xy...

>“= { ...x~y...}~-~-

STRONGLY

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ALGEBRAS

149

{ . ..yx...}. to movey’s to the left ofxEX (noting that X(“)oyc&‘(“)c A?@-‘) since Xc”) remains an ideal in 3 by (0.14)). From (2.3’) we get {y,...y,h(x,, .... ~,)y,+~..~~,,} =h’(x ,,..., x,, y,, .... y,) E X(Xu Y) a Jordan polynomial homogeneous of degree 1 in each yi; if we specialize yi to ai, xi to bj in J then when one of the ai falls in the ideal Z so does the whole Jordan product h’(b, , .... b,, a,, .... a,) involving a,. This is ideal absorption (2.4). For envelope absorption (2.5) we note the general result (2.7)

if ZC .Z has ZoJc I (e.g., if la J) then for any associative envelope A of .Z we have yA(Z) = Zj = AZ

since~~(Z)=AIZAI=~~Z~m=~~Z(usingzx=z~x-xzforz~x~Z~JcZ to move XEJ to the left of z E I), and dually. Thus we can write a=Cx, . ..x.z for x,E.Z, ZEZ. Let Z(a)=max{m+l}kl be the maximum of the lengths of monomials appearing in a (so UE?Z), and set n(a) =21(a). When a = a* E H(A, *) n yA(Z), t E SCfl), then {x a t} = xat + ta*x is spanned by the xx, . ..x.,,zt + tzx,...x,x = {xx1 .-.x,zt} c .-*\ {JJ...JZ@“) U,t=a*ta

c In 2’ by (2.4) since n > 2(m + 1) 2 m + 2; similarly is spanned by zx, ‘..x, tx-‘-xmz= U,U+--- U tc U,s@“‘cZ n XC”) and by zx, . . . x, tx’, .. . xkz’ + z’xk, . . . x; fi, . . . x,7 =

,m+‘, . . . J Y?(“) i.yyJ Z} c In 2 by (2.4) since i zxm “‘X, tx; . ..&.Z’)E{J n>2(m+ l), n32(m’+ l)=sn>m+m’+2, so we have (i). For (ii), if a E t(YA(Z)) is spanned by elements z E Z and traces

tr(x, “‘x,z)=(x,

..-x,z)+(x,

. ..x.z)*=

then U,a is spanned by U,Zc U,Zcln~? ,m+l, {JJ ... .ZZ%““‘}cZn%

(x1 . ..X.Z}

and by U,{x, . ..x.,,z} =

by (2.4) since n>m+2. Thus { tx1 -x,zt}E (ii) holds in this case, and such a lie in the absorber. In (ii) the particular case a= {xl...x,z} is tracial with I(a)=m+ 1, n(a)=2m+2, yielding (iv). For (iii) we only need to show

which by (2.7) follows since L. = Hn YA(Z) is spanned by b = b* = C aizi (ZiEZ, ai=x, “.x, for xioJ), so (1jL) is spanned by xyb+b*yx= and U,j is spanned by b*xb= xiziaf xaizi+ C tr(x.Wizi) E t(A(O) ~i,jtr(zia,*xujzj) for zia*xaiz,=U,U,_-..U,,xEUIUJ...U,JcZ and

150

MC CRIMMON

AND

ZEL’MANOV

tr(z,a*xujzj)Et(9A(Z)). From (i) and (ii) we see that t(&(Z)) lies in the union of absorbers. Universal absorption (2.6) is the special case A =a(Xu Y), Z=.Z= f(Xu Y) of (2.5). [ 2.8. Remark. For simplicity we have only required a hermitian ideal to be formal, but for all practical purposes we could have required it to be a linearization-invariant T-ideal. The reason is that we are interested only in the values X(J) which the ideal assumes on Jordan algebras J, and X’ and its T-closure L% (the smallest T-ideal containing #, namely the set X(f) of all substitutions in &‘, i.e.: all values assumed by 2 on the universal d) both assume the same values on any particular algebra,

A?(J) = 2q.z). Moreover, the actual values assumed by &’ are less important than whether or not they are all zero, and we give up hope that .Z has a nonzero hermitian part only when all hermitian polynomials vanish strictly on .Z, 2(J)

= 0 strictly 0 P(J)

= 0,

for 2’ the ideal of all linearizations of 3. Thus it is also natural to restrict attention to ideals &‘= 9’ that are closed under substitution and linearization. Note that if S is hermitian in the sense of (1.2) then so is its T-closure 3: the elements of 2 have the form h(p,, .... p,) where pie 9 and so any n-tad in 3 has the form {h,(p,,, .... plm), .... 0 i, .... x,)EX, hn(p nl,...,~nm)}=h(~ll ,... ,~~~)~~‘,whereh(x,,,...,x,,,...,x,~,...,x,,)= ~11, .... x~m), .... h,,(x,~, .... x,,)} E 2 by choosing disjoint sets X, = xkl, .... xkm} in the infinite set X and using formality (1.1) to replace the ihl’ variables in hk by those in X,.

Also, if S is hermitian so is its linearization-closure L%?’ (the set of all linearizations, equivalently the set of all coefficients h, of fixed monomials te = t;\l . . . t$; in h(x+t~y)=h(x,+tllyll+~~~+tl,y,,,..., for he X, T an infinite set of scalar xm + tm1 Yml + *. . + t,, yrns)~ f[T] indeterminates and {xi, yV} distinct indeterminates in X. Once more, 2’ is closed under n-tads since the n-tad {h!:) a.. hg)} is the coefficient of t(lh . . . t’n’en in {h(‘)(x(‘) + t(l). y(l)), .... h(“)(x(“)+ t(“) . y’“))} for { t$F)} distinct in T and xl’), y$,!) distinct in X. Thus we may always enlarge a hermitian ideal to obtain one which is closed under substitution and linearization. Invariance under substitution

STRONGLYPRIMEQUADRATICJORDANALGEBRAS

151

is just invariance under all (algebra) endomorphisms of $, and invariance under linearization implies invariance under all derivations of 9 by (2.9)

w4x,,

for d,Y:p the linearization )lyi,

.*3

-**,x,)) = f: Llpp i=l

obtained

as the coefficient of 1 in p(x,, .... xi+

xn).

In fact, by [373 one can choose X(X) to be homotope-invariant, in the sense that if p(xi, .... x,) E X(X) then #“)(x1, .... x,) E %‘(Xu {y>), where P’y’(x, , .... x,) is the value ofp on the elements xi, .... x, in the y-homotope Y(Xu {YIP (1’ivm. g inside a’-“) with associative product a . ,b = uyb). For some interesting applications of n-tad eating (2.3’) see [ 11. 1

3. SPLITTING SEMIPRIME ALGEBRAS INTO ANTI-HERMITIANS AND SPECIAL HERMITIANS Here we show how an i-special semiprime algebra splits as a subdirect sum of special prime algebras of hermitian type and prime algebras of antihermitian type. The i-special prime algebras are themselves either of special hermitian or of anti-hermitian type, so the existence of a nonzero hermitian part forces a prime algebra to be special, and hence a hermitian form. We say an i-special algebra J is of anti-hermitiun type if all its hermitian parts vanish, X(J) = 0, equivalently

if all hermitian

hermitiun type if its hermitian

polynomials vanish on .Z. We say .Z is of parts are “dense” in the sense that they

cannot be annihilated, = n Ann,(X(J))=O. JP Note that a prime algebra has hermitian type iff it has some nonzero hermitian part z(J) # 0. An i-special algebra J= S/Z is the homomorphic image of a special Jordan algebra S c ZZ(A, *) (S generates A ) by an ideal Z u S. The natural specializing ideal of J with respect to the data Z, S, A is (3.1)

SP I, s,A(J) = VZ

(f= Sn YA(Z)).

152 Dividing

MCCRIMMON AND ZEL'MANOV

out by this ideal turns J into a special algebra, the natural

specialization

8, s, A(J) = JISP,, s. A(J) 2 S/I (Note that S/r=S/SnYA(Z)z (S+YA(Z))/9JZ)cA+/YA(Z)=A+ for A an associative algebra.) There is always a smallest specializing ideal Sp(J) with Y(J) = J/Sp( J) special, so here Sp, s, A(J) I Sp( J). 3.3. HERMITIAN DICHOTOMYTHEOREM. For any i-special Jordan algebra J and any hermitian ideal X(X) a f(X), the ideal X(J) n Sp(J) lies in the Baer-radical, so ty J is semiprime then

where H is special of hermitian type (a subdirect sum of special prime algebras of hermitian type) and H’ is of anti-hermitian type. Zf J is i-special and prime, then either J = H is itself special of hermitian type, or J= H’ is of anti-hermitian type. In particular, a prime i-special algebra J having a nonzero hermitian part s(J) # 0 is necessarily special. Proof Since Sp( J) c Sp, s, A(J) and any subalgebra of a Baer-radical algebra is Baer-radical, it suffices to show that X(J) n Sp,, ,.,(J) is Baer-radical. Here by (2.7)

YA(Z) = AZ= za

Va

J)

consists of all a = x ai for ui= s1 . ..s.z (SUES, ZEZ) as usual, so f= S n YJZ) has P c U,, x”(I) 3 and hence by absorption (2.5iii) m f3 = u Abs,(/i, e), s(Ws) (n)>0 II=0

Then S’n13cXn(U Abs,,,,.,.(&‘“),Z)) (2=%(S)) ciJ;=,Xn Abs, s(%(“), I). But each L, = X n Abs, s(~(m), I) is solvable mod Z, so LJ lrn+‘) = U,O#,(“) c Ux(,) (Abs, s(#(m), I)) c Z (by definition, D(2PnT)=(SnIjJ cXnT3~UL,,,cW,(S,Z) and %nfc&$(S,Z). Thus in J= S/l we have WJ)

is very Baer-radical.

n SP, s. AJ) = g*(J)

(Recall that g(J) = U gA(J) for

~A+l(J)=~

{Illa

J,fin)~9Y~(J)forsomen}.)

STRONGLYPRIMEQUADRATICJORDANALGEBRAS

153

If J is semiprime then &‘(J) n Sp 1,s, A(J) = 0 for all 2 and all presentations J= S/I. If J is prime then s(J) n Sp(J) = 0 forces either X(J) = 0 or Sp(J) = 0. Thus either J= H’ is anti-hermitian (all X’(J) =O), or else X(J) # 0 for some ~9 and Sp(J) = 0, in which case J = Y(J) = H is special with 2(J) # 0, therefore by primeness Ann,(X(J)) = 0, and H is special of hermitian type. Coming back to the semiprime case, we have Jz n J, for prime images J,; these images remain i-special, so are either of anti-hermitian or hermitian type (some H7(JY) # 0). Let H’ = n J+ be the product of the antihermitian factors, and H = n J, the product of the hermitian factors. Then fz H q H’, where H’ itself is anti-hermitian and H is of hermitian type (%?(J,) # 0 => AnnJY(Z7(J,)) = 0 by primeness, so z E n, Ann,(X(H)) Zzy E Ann,(ZY(J,)) = 0 for all y => z = 0. 1 We will see in Part II that anti-hermitian algebras are inherently “degree 2.” Once we have the speciality part of 3.3 we can immediately extend the Special Hermitian Structure Theorem 1.6 to the general i-special case. 3.4. HERMITIAN STRUCTURE THEOREM. An i-special algebra which is prime and hereditarily semiprime and has a nonzero hermitian part X’(J) # 0 is a hermitian form

for a *-prime associative algebra A with Martindale *-algebra of quotients Q(A). I PART

II. ALGEBRAS OF CLIFFORD TYPE

We have seen that prime i-special algebras J which are not hermitian are anti-hermitian: all hermitian polynomials vanish. Later we will see that one of these hermitian polynomials is a Clifford polynomial, which does not vanish on 3 x 3 hermitian matrices. In this part we make a general study of algebras of Clifford type, those satisfying a clifford identity. We carry out the study for nondegenerate algebras, although it would be sufficient to work with strictly-nil-free algebras since in fact any nondegenerate algebra satisfying a polynomial identity is automatically strictly-nil-free. The i-special nondegenerate algebras of Clifford type satisfy the standard Clifford identity, and the prime ones are forms of Clifford algebras J(Q, 1), and so in particular are special. The structure of Clifford forms in characteristic 2 is vexing, and requires a surprising amount of work proving things that are obvious in characteristic 22.

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AND ZEL'MANOV

4. CENTRAL CLOSURES

We begin by showing that a prime algebra can be “centally algebraically closed,” i.e., has a scalar extension whose centroid is an algebraically closed field. This is easy for linear algebras, but laborious for quadratic algebras. Since free scalar extensions do not suffice, we work with tight ones. We begin by recalling that we can always shrink the Jacobson radical into the nil radical by a polynomial extension as in (0.4’). 4.1. PROPOSITION [ 15). (i) Zf J is strictly-nil-free than the free scalar extension J[T] = JO* @CT] (or, zf @ is a field, J(T) = JO@ O(T)) is semiprimitive for any infinite set T of scalar indeterminates. (ii)

Zf CDis a field, then for any extension field b I @ which is big, IfiI>l+dim,J,

and any scalar extension j= dJ, we have Rad(J) = Nil(J), it is semiprimitive. 1

If J is nondegenerate centroid (4.2)

f(J)=

so if? is nil-free

then J may be considered as an algebra over its

{TEE~~,(J)ITU,=

U,T, UTX= T2U,forallx~j}.

For prime algebras the centroid T(J) is a domain we can pass to the central closure I=

r-‘J

acting faithfully on J, SO

(r= f(J)).

Note that if T(J) is already a field then J= J is already centrally closed. Unfortunately, in the quadratic case if we try to pass to J= FJ for r the algebraic closure of Z, 3 may in the process have grown a bigger centroid than ?; so we must again form a central closure, leading to an infinite chase. Luckily this chase comes to an end: we can find a closure whose centroid is an algebraically closed field. 4.3. CENTRAL ALGEBRAIC CLOSURE THEOREM. Zf J is a prime Jordan algebra, then for any field Q =) T(J) there is a prime tight unital-scalar extension 7 = 03 whosecentroid r(J) = fi is an algebraically closedfield 8 3 52.Zf J is strictly nil-free, then d can be chosenso that f is semiprimitive. Proof

(4.4)

We can always pass to a central algebra over a field,

If J is prime then its central closure J= T-‘J is a tight cover of J and is a prime algebra over the field r= T-‘Z’. If J is

STRONGLY

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QUADRATIC

JORDAN

ALGEBRAS

155

unital then the centroid r(J) is a purely inseparable extension of F, T(J)*cF’cr(J), Ir(J)I
If J is an algebra over a field CD,then its unital @hull J is a tight cover of J.

Indeed, if J is already unital then J= J is very tight; otherwise J= @l @J as in (0.6), which is also tight: if al + x#O in ia .? with in J= 0 then since @ is a field we can divide by a and see that i contains an element l-u with UEJ, where for XEJ, U,(l-u), x0(1-u), tIpuxEfnJ=O shows x2 = UXu, 2x = x 0 U, U,x = u 0 x - x = x, so u is a unit for J, contrary to hypothesis. (Note this result fails for J over a domain ~0: J= 22 is prime without unit over @= Z, but its unital hull J= Z1$22 is not tight, because E(20 -2)n J=O.) It is easy to pass to any field extension: we take a free scalar extension and tighten. (4.6)

If J is an algebra over a field 0 and IR 3 Qi any extension field, then J’ = 52J= (JO, Q)/M is a tight sclar extension of J for any ideal M Q JO@ Q maximal with respect to MnJ=O.Here IJ’I
Indeed, by construction J’ is a tight cover. In general I JO, Q I
156

MC CRIMMON

AND

ZEL’MANOV

simple we must make sure @ is the whole centroid of J, and even then in characteristic 2 we will see in 5.2 examples of central simple algebras which do not remain even semiprime under scalar extension, such as J(Q, 1) for Q(a,@a,@a,)=a~+a~o,+a~o, in J=@l@@x,@@x, for 1, oi, o2 E @ independent over @‘.) Now we combine these to centrally algebraically close a prime algebra. We start from a prime J and form J= r- ‘J over r(5) as in (4.4), then adjoin a unit to get J= r(J) .? by (4.5). If weCwant to include some general Q we next form a tight scalar extension BJ= Qj as in (4.6). Even if we have no 52 in mind, we perform this step to make sure that our base field is infinite. The resulting algebra remains prime by tightness, but may have centroid bigger than 52, so we apply (4.4) once more to get a tight scalar extension (*)

J() = @().I,

Q0 infinite.

We now try to algebraically close the centroid. As above, we will pass to the algebraic closure So by (4.6), then centrally close by (4.4), and repeat this process over and over, using the fact that a tight extension of a tight extension remains tight. Starting from Jo we recursively construct for each ordinal A a tight scalar extension (**)

whose centroid contains the algebraic closure of the previous ones. We have delined this for 1= 0 in (*). For a successor ordinal 1+ 1 we use (4.6) to pass to the algebraic closure JAo = $A Ji = Si,j and then (4.4) to pass to the central closure JA+ I =G over #A+ i = f(J,+,)I>T(J,,)I>~,.Here IJA+l I 1< I Jl,, I (by (4.4) and infiniteness) = I Jl I (by (4.6)) = I J, 1 (by the induction hypothesis (**)), so 1Qn+ I ( < 1JA+ , ( = 1Jo I. Thus (**) holds for I + 1. For a limit ordinal I we take JA,, = l&, < 1 J, = U, <1 J,, which is a tight cover since each J,, is, and then use (4.4) to take the central closure JA = z = @AJA,, = @,f for Gi = r( J1) I r( JA,) I @,, + 1 I 6’a for each p < ;i (since then also p + 1~ 1 for a limit ordinal A). We have 1G1 I < I JA I = 1JAo 1 (by (4.4) and infiniteness) = lim, < 1 1J,, 1= lim, < 1 1Jo 1 (by induction hypothesis (**)) = 1Jo I. Thus (**) holds for 1. Now the increasing chain of fields Qi must come to a stop: there is a & with @L+1= @& (hence by induction @A= @& for all il a&,), since otherwise @A+,>@,?

for all 1

*

l@,+,l~IIlforall~,

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which by (**) is impossible for ordinals I with 1II > 1Jo). (If for each i there is tll E @A+ i\Qi then the ai are all distinct, so 1QA+, 1> I{a,~~~1}~=~{~~~61}~~~1~,) Then s=J,, a=~@~,, meets the requirements of the proposition: as a tight extension it inherits primeness, and @ti=@no+l~5,, implies s’i = r(J) is algebraically closed. If J is nilfree then so is 3 by (0.17); if we want it to be semiprimitive we just start with a large Sz in (*) and apply 4.l(ii). 1 In the next section we will see an example where the central closure is not straightforward.

5. SCALARS IN CLIFFORD ALGEBRAS

In Section 7 we need to know that certain elements in Clifford algebras which behave like scalars are (essentially) scalars. In linear Jordan algebras these elements are true scalars in the center, but strange things happen in traceless Clifford algebras of characteristic 2. For a quadratic radical to be

form we define the (quadratic) Rad(Q)= Bilrad(Q)=

{z~JlQ(z)=Q(z,

radical and bilinear

J)=O)

{zEJI Q(z, J)=O}.

These two radicals coincide if 4~ @, but not in general. We say Q is nondegenerate if Rad(Q) = 0, and anisotropic if a(z) = 0 *z = 0. Over a field we define the defect of Q to be dim, Bilrad(Q), and say Q is defective if it

has defect > 1. Since Rad(Q) = {z E Bilrad(Q)l Q(z) = 0}, nondegeneracy can hold only when Q is anisotropic on Bilrad(Q); over an algebraically closed field this can hold only when the defect is Q 1, so nondegenerate Q over algebraically closed fields are nondefective. 5.1. PROPOSITION. Let J = J( Q, 1) be a Clifford algebra of a nondegenerate quadratic form Q over a field @. (I)

Zf @ has characteristic # 2 the following are equivalent for an element

CEJ:

(i) (ii) (iii) (iv)

all [c, x, x] for XE J are non-invertible; all Cc,x, x] are nilpotent; either (a) CE @l, or (b) dim J<2,

[c, J, J] = [J, c, J] = 0.

[J, J, J] =O;

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MC CRIMMON

(I’) Zf CDhas characteristic

AND

ZEL’MANOV

2 the following

are equivalent for an element

c E J:

(i) (ii) (iii) (iv) (II)

all c 0 x for x E J are non-invertible; all c 0 x are nilpotent; either (a) c~@l,

or (b) dim JG2,

Jo J=O;

Zn all cases, if such c is nilpotent then c = 0.

Proof:

Recall the operations (0.9) in J(Q, 1): xi = T(x) x - Q(x) 1

(1)

XOY = T(x) Y + T(Y) x -

(1’) (2)

(a’) c, 1 ~Bilrad(Q),

coJ=O.

U,Y =

Q(x, J) x - Q(x) Y

Qb, Y) 1 (Y = T(Y) 1 -Y).

Over a field we have

(3)

z nilpotent

o z2 = 0 o T(z) = Q(Z) = 0.

For associators [x, y, z] = (x 0y) 0 z - x 0 (y 0z) we have

(5)

+ {~(z)Qtx,~)-~(x)Q(~,z)) 1 Cc,x, xl = {T(c) T(x) - 2Q(c,x,} x - ( W2 - 4QW) c

+ {T(x) Q(Gx)

- WC)

Q(x)> 1.

For (i)o(ii) in both (I) and (I’), since all [x, y, z] (and, in characteristic 2, all x 0y) have trace zero, by (3) they are nilpotent iff they have norm zero iff they are non-invertible. (iiia)*(iv) and (iiia’)* (iv) are clear (using (l’)), as is (iv) 3 (i). To see (iiib) * (iv), if J= @l + @u has dimension < 2 then [J, J, J] = [Gu, @u, @u] = 0 (and in characteristic 2, Jo J = Qiu 0 CPU= 2@u2 = 0). Thus (ii) =E-(iii) if dim J < 2. From now on we assume dim J> 2, and show (ii) * (iiia) or (iiia’). First consider the characteristic 22 case, where Q(x, y) is a nondegenerate bilinear form, and T( 1) = 2 # 0. To show c E @l it suffices to show (6)

T(c)=O=c-c=o

since then c= al + c0 for T(c,) =0 forces co=0 (by (6), [c,, x, x] = [c, x, x] = 0) and c = crl E @l as in (iiia). We have T(c)=O*Q(c)=O

noting

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159

since otherwise J= Qjl I @c I V for V# 0 by dim J> 2, hence Q(V) # 0 (else Q( V, V) = 0, Q( V, J) = 0, VC Rad( Q) = 0 by nondegeneracy), and if XE V had Q(x) # 0 then T(x) = Q(x, c) = T(c) = 0 would give [c, x, x] = -4Q(x) c invertible by (5) (Q([ c, x, x]) = 16Q(x)’ Q(c) # 0). We claim Q(c)=O*c=O, since otherwise J= @l I {@c + 4U) I V for a hyperbolic pair (c, d}, and if Q(c)= Q(d)=O, Q(c, d)= 1 then for x=c+d Q(x)= Q(c, x)= 1, T(c) = T(x) = 0 would give [c, x, x] = -2x + 4c = 2(c - d) invertible by (5) (Q(2(c - d)) = -4 # 0). This finishes (6). Now consider the characteristic 2 case, where T( 1) =0 and Q is anisotropic on Bilrad(Q) by nondegeneracy. We have necessarily T(c) =O,

since otherwise J= (@lo Cpc) I V relative to Q(x, y), where again some XE V would have Q(x) #O, so by (l’), T(x) = Q(c, x) =0 would give c 0x = T(c) x invertible (Q(c 0 x) = T(c)* Q(x) # 0). If T vanishes identically then 1 E Bilrad(Q), and we claim CE Bilrad(Q) too as in (iiia’): cox = Q(c, x) 1 noninvertible => Q(c, x) = 0 for all x * c E Bilrad( Q). If T does not vanish identically, T(u) = 1 for some U, we claim c E @l as in (iiia): J=(~l~~u)IV,andc=al+c,for~~~V,~oitsuff~cestoprovec,=O (note c,, 0J = c 0 J = 0 too), so we may as well assume from the start that T(c) = Q(c, U) = 0; then by (l’), COu = c noninvertible forces Q(c) = 0, and cox=T(x)c-Q(x,c)l noninvertible forces 0 = Q(cox) = Q(c, x)*, whence all Q(c, x) = 0 and Q(c) = 0 force c E Rad(Q) = 0. The property (II) is clear from inspection in (iii): in (iiia), c=al nilpotent forces CI= 0; in (iiia’), c E Bilrad(Q) nilpotent forces Q(c) = 0 and c = 0 by anisotropy; in (iiib), J of dimension ~2 has no nilpotents at all (it is a division algebra or a split @e, @ @e,; alternately, if Q(z) = T(z) = 0 for z # 0 then J= @l + @z, Q(z, J) = Q(z, @l + @z) = 0 = Q(Z) forces ZE Rad(Q) =O). 1 5.2. EXAMPLE. To show that the central closure (4.2) in characteristic 2 does require some delicacy, consider the case J= J(Q, 1) for a defective nondegenerate form Q over a field 0 of characteristic 2, so the bilinear radical has dimension > 2. We noted that nondegeneracy means anisotropy of Q on B = Bilrad(Q), and this is easily upset by scalar extensions: over an algebraically closed field d the only anisotropic spaces have dimension < 1, so Ba is no longer anisotropic, and the free scalar extension Ja = J(Qa, 1) has grown a radical of dimension 2 dim, B - 12 1. Thus even when J( Q, 1) is central simple over @ (as is the case if dim, J/B 2 3) the free scalar extension Ja need not be semiprime. Thus we cannot hope to use free extensions

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AND ZEL'MANOV

in (4.2). But why does dB not cause us trouble in the tight scalar extension of (4.2)? If we write J= B@ I/ for a vector-space complement V of B in J, we can easily show that fiB = Bilrad(Q) in dJ= J(& 1). However, in the non-free extension this space B of @-dimension 22 must be collapsed to a space fiB of a-dimension 1 to preserve nondegeneracy of 0. Indeed, the mapping B + e d is a @-linear imbedding of B into a field d containing a copy d of @ (via E = a’), so we have J(Q, l)=B@VGSlil@Vn=J@,

1)

for Q(Gl @C ~5~0 vi) = ~7’ + C G~Q(ui)* + C c~@~Q(u~, Vi)’ quadratic over d (note Q(i(6@ u)) = &Q(6) 10 (10 u)) = Q(b)* + Q(v)* = i(Q(b) + Q(u)) = i(Q(b@ u)). In effect, the bilinear radical B is really part of the scalars, not part of the quadratic form.

1

6. CLIFFORD FORMS

In this section we give an intrinsic description of Clifford forms. An important property of albert forms is that their central closure is already an albert algebra. The analogous result fails for Clifford forms in characteristic 2: if the algebraic closure is tiny or small of dimension 1 or 3, then the central closure need only be amply sandwiched between two Clifford algebras, while if the algebraic closure is large of dimension 24 then the central closure itself is a Clifford algebra. Smallness or largeness depends on the dimension of the algebraic closure; in the central closure this distinction is represented not by dimension, but by certain polynomial identities. Throughout this section we are concerned with Jordan algebras J which are forms of simple Clifford algebras 7 = J( 0, 1) of quadratic forms over an algebraically closed field fi : fif = 1. Simplicity of 3 forces nondegeneracy of the quadratic form 0, and algebraic closure forces the defect of Q to be < 1. Since 3 is simple we know J is prime, and hence has a central closure J= T(J) - ’ j which is prime with centroid 6 = r(f) a field. Our goal is to show that J is (almost) a Clifford algebra. Recall our notation JO-3 J of (0.13’) for ample outer ideals. 6.1. CLIFFORD FORM THEOREM. If J is a @-form of a simple Clifford algebra f = &j= J(e, 1) ouer an algebraically closed field 0, then the central closure J= r(J)-’ 3 is simple ouer the @-field 5 = r(J), and either (I) Jiela3,

dim J= 1: f is a division algebra amply sandwiched between two

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161

for~cdc~c~(so/=~+=J(~,l)ifchar#2),whereforanyunital commutative @-algebra D we have D+ = H(D, *) = J(Q, 1) both hermitian and cltfford for * the identity involution and Q the quadratic form Q(d) = d2 (for J itself we have 12: +I J= 521 + $2: for commutative associative domains Q,, 9, c Q c Q,, LI: c $2,);

(II) dim ? = 3 : J is amply sandwiched between two clifjford algebras, 1, +I .i -+ J, for Ji = J( Qi, 1) cltfford algebras of nondegenerate quadratic forms oi over fields si, 8ic G-,c *I = &$~a (so J=JO= J(p, 1) is chfford $ char # 2); or (III)

dim J” 2 4 : J = J( Q, 1) is a Clifford algebra over 8.

Since all the difliculty is in characteristic 2, let us first settle the characteristic 22 case: / is Clifford over 6 since the values attained on J by Q lie in 6. Indeed, it suffices if the values of p lie in 8, since O(X) 1 = F(x) x-x* shows 2&(x) = T(x)* - p(x*), and we can recover & from ?=in characteristic # 2. If dim ?= 1 then J= Jn 01 = $1. Linearizing [x, y]* E& shows [a, b] 0 [a, c] l 3n dl = $1 for a, b, ce.7, and if dim J> 3 then in characteristic 22 there exist [a, b] 0 [a, c] # 0; then [a, b] 0 [a, x2] = ?‘(x)[a, b] 0 [a, x] shows T(x)E 8 unless [a, b] 0 [a, x] = 0, in which case [a, b] 0 [a, x + c] # 0 shows F(x + c) E 5 and F(c) E 6, so in all cases F(x) E 5. From now on in this section we will assume CHARACTERISTIC

= 2

and hence will dispense with all minus signs. A simple Clifford algebra 3= J(& 1) over an algebraically closed field s”i has either dimension 1 or dimension 23 [the 2-dimensional case is ruled out by simplicity: the bilinear radical must be zero, so a proper idempotent e, can be found, hence f = be, EBde,, contrary to simplicity]. We say J= dl of dimension 1 is a tiny Clifford algebra; it satisfies [x, y]* = 0 and [x, y] z x oy = { xyz} = 0, &J, J”) = 0. We say 7 of dimension 3 is a small Clifford algebra; it satisfies [x, y]* f: 0 but [x, y] 0 [x, z] = 0 and &(J, {~~~}) = 0 (here JOT= {m} =s”z u is the bilinear radical of e). J” is at the same time an isotope of a hermitian algebra: if 7 is traceless we can find an isotope J@‘)= J( @‘, 1‘“‘), l(U) = v - 1 uy= U,U”, Q(“’ = Q( 0) (7,

F”’ = F( v, . ),

with nonzero trace, hence JcU)= fie, @ du 0 liie, is split, ?(‘) z H(M,(W)) via &,e, + Gu + G,e, + (3 $). With hermitian algebras we often have to settle for ample subspaces, so it is not surprising that for small Clifford

162

MCCRIMMON

AND ZEL'MANOV

algebras we will have to settle for ample outer ideals. We say 3 is large if it has dimension 24. In this case [x, y] 0 [x, z] & 0, e(?, 70 J) # 0. These properties are reflected in forms J of 2 (subalgebras having I= fiJ as unital-scalar extension). We say that J is a tiny, small, or large Clifford form if it is a form of a tiny, small, or large Clifford algebra 7. Note that the size of a form is not measured by dimension: a centrally closed form need not have dimension 1 to be tiny, nor dimension 3 to be small, because dimension can shrink in the passage to J (elements which are independent over the centroid in J may be dependent over d in 7). Size is instead measured by identities and by behavior under 0: a Clifford form J is (6.3)

(I) tiny~[x,y]-0oJ0J={JJJ}=&J,J)=0 (II) small-=-[x,y]’ f 0, [x,y]o[x,z]=Oa &(J,J)#O, Q(J, {JJJ},=O, JoJfO (III) large 0 [x, y] 0 [x, z] + 0 0 Q(J, Jo J) # 0

since a multilinear identity holds for a form J iff it holds for j: It is important that forms of all sizes are algebraic (though further removed from being degree 2 than in (10.4)).

one step

6.4. ALGEBRAIC LEMMA. Let J be a form of ?= fij = J( 0, 1) which is central over a field 0 of characteristic 2 (I’(J) = CDId). Then

(i) (ii)

Jnfil

=Q,l

for Q,cfi

with sZ$c@;

J is algebraic over CD of degree <8: x8 + ax4 + /I1 = 0 for a = F(x)4, /?= Q(X)4E @;

(iii) (iv)

x is invertible in Jox

is invertible in 10 e(x) # 0;

x~~@~~x~E~~csx~E~~~~(~)=O.

Proof: (i) Jn al= Q,,l for some @-subspace Q, of 6, and QiId = u QOlcr(J)n U,cf(J)nEnd,(J)cT(J)=@Id, so sZic@. (ii) If dim J= 1 we have x2 E @l by (i), so assume dim J> 2. From x2 + p(x) x+ e(x) 1 = 0 in the clifford algebra j over 0 we get x8 +ux4 +pl =0 for tl= !Qx)~= T(x”), /3= Q(x)~= 0(x4) in d (note xi 0 xk = 0 and T(T(Y)’= T((y’) in characteristic 2). If x4 E s”il then x4 = o,, 1 by (i), where both a = T(x*) = 0 and fl= &(x4) = ~8 lie in @ by (i). On the other hand, if x4 4 dl then the tl, B in (ii) are uniquely determined; we will show there exist a,,, &, in @ satisfying (ii), so c1= ~1~and fl= /I0 lie in @. Now x4$& =x$dl u Bilrad(& (such an x would have p(x)=O, x2=&x) 1~61), so by 5.1, dimJ>2 implies O#(XOJ)~=[X, J12c Jnfil =52,1 (by (i)), hence some [x, y]‘= w,,l ~0. Then by (10.2’), ~,x~+~~x~+v~l=O for o,,~O,~O~QO, so x8+a0x4+/&1=0 for q,=o,y2&, &=o;~v~ in sZgc@ (by (i)).

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(iii) x invertible in J*x invertible in s (or any unital cover) * &x)#O in J(e, l)=~.j=&x)~#O in @ax has inverse x-l= pp’(x7+tlx3)~J by (ii). c@l (iv) ~(x)=O*x*=Q(x) 1 EBl *x2~SZ01 (by (i))*x4EQil (by (i)) * x8 E dl = T(x)’ = F(x”) E T(fil) = 0 = F(x) = 0, using the fact that F(y)* = p( y*). u The Clifford Form Theorem 6.1 in characteristic 2 will follow by applying the following theorem to the central closure J over 8. 6.5. CENTRALLY CLOSED CLIFFORD FORM THEOREM. Let J be a unital centrally closed Clifford form of characteristic 2: T(J) = CDis a field, and J”=~J=J@, ijf or a nondegenerate 0 over an algebraically closed 8. (I)

Zf J is tiny, then J is a division algebra sandwiched between two fields: @+ -3 J=1;21 -3 @+.

(II)

If

J is small, then J is amply sandwiched between two Clifford

algebras

nondegenerate quadratic forms Qi over fields Qi with @c QO= @[~(J,,)]~@,=@@(J)]~fic8.

for

(III)

Zf J is large, then J= J(Q, 1) is Clifford over @.

Proof: (I) We begin with the case where J is tiny, 3= dl. Then J= 521 for some Jordan @-algebra Q cd. Define

(Ii)

is,=

{oEQ;21wSZca}

(Iii)

52, = @[Q] = subalgebra of B generated by 52.

Then Q, c Q c Q, for domains Qi, and 0: c J c Sz: . Here Sz: c 9, since this is true of its generators ~~52: 02Ql= U,iQl c U,Jc J=Ql. So far we have made no use of central closure (as in 6.1(I)). Now assume J is centrally closed: then @ c Sz, c { cZE d )dX2 c Sz} c T(J) = @, so 52, = @, hence (Iiii)

fqc@

shows that L2, is a field contained in J @. Since all nonzero elements have invertible squares, Q2 c @ by (iii), we see that Q is a Jordan division algebra, and trivially in characteristic 2 (iii) implies ampleness @-+l2-+/5 (III) Next we consider the large case. We must show that the values bO7/69/2-3

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taken by Q on J all lie in @, so that the restriction Q = Q 1J is a Q-valued quadratic form with J = J(Q, 1). Since { & E d 1GJc J} is contained in T(J) = @, it suffices to prove (IIIi)

&J)JcJ.

By largeness (6.3111) there are a, b, CE J with (IIIii)

[a, b] 0 [a, c] = &l # 0

in J,

so by 6.4(i), 0 # 6’ = LYE @. Now in general (IIIiii)

F(z) = 0 =s F(J)(zo

(2EJ)

J) c J

since ~(x)(z~y)={~~z+~(z)~+~(~,x)1}~y=(x+y+O+OcJ (p(z) = 0 = 2). Thus in particular for z = [a, b) = a0 b we have aT(J) 1 = T(J)G21 = ~(J)(ci3[a,b]o [a,c]) = F(J)([a,b]o {a,Gl,c})c F(J)(z~J)cJ by (ii) and (iii), and since c1is invertible in @ we have (IIIiv)

T(J) 1 c J,

J=J

(here the bar indicates the Clifford involution, not central closure). The same holds for all isotopes J(‘) (since they remain large Clifford forms), so J=J p”(J) l’“‘= C&J, u-l )&)]u-‘=&(J, U;‘u) U=&U,lJ,u)ii = Q( J, U) ii; replacing u by iz (using (iv)) we get (IIIV)

&J,

Then by (iv), (v)), so

&(u)J=&(u)J=U,J+B(U,J)UCJ+~(~,J)~=J

(IIIvi)

U) u c J.

(by

Q(U)JCJ

for invertible U. Thus Q(x) JC J as in (i) when x = u is invertible, and if x is not invertible in J then Q(x) = 0 by 6.4(iii) so trivially Q(x) JC J. Thus (IIIi) holds in all cases, and J is Clifford over @. (II) Finally, assume that J is small, so by (6.311) (III)

&J, {JJJ})=O

(IIii)

T({JJJ}

(IIiii) In place of (IIIi)

Jo J#O.

we have the weaker relations

(IIiv) WV)

=0

Q(J){JJJ) &J)2

JC J

c (JJJ)

(equivalently

c J. &J)2

c @)

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Indeed, for (IIiv) we have { yzw} = { yzw} from (IIii), hence Q(x){ yzw} = U,(yzw> + 0(x, { yzw}) x = UX{ yzw} + 0 (by (Iii)) c (JJJ} since this is an outer ideal in J. For (11~) it is easier to pass to a traceless isotope. By (IIiii) we can choose 0 # u E Jo J, so by (III), u E Bilrad(Q) and therefore by nondegeneracy of Q we have Q(u) # 0 and (by 6.4(iii)) u is invertible. The isotope J(‘) (as in (6.2)) is traceless: F”(x) = &(x, U) = 0(x, u) = 0, so by 5.1(i) squares are scalars x (2,“) = Q(‘)(x) 1(“. Therefore d Id 3 Q(“)(x)* Id U$!” = U$‘)’ = U, U, U, U, E End,(J) by (6.2) so @‘)(x)* E T(J) = @ by Lr hypothesis of central closure; thus by (6.2), (Q(x) Q(u))* E @, so x = 1 shows Q(u)~ # 0 in @, and therefore all Q(x)’ E @. (Another indication that traces are crucial is that for (11~) it s&ices if (IIV’)

T(J)” Jc J

or

(IIV”)

T(X)2XEJ

since modulo J we have 0 = Ux2 y = UR~).~+ pcxjl y = T(x)’ U, y + implies Q(x) xoy + Q(x)’ y s U,{ F(x)’ y } + Q(x)* y (since (IIiv) %) as well), so (11~‘) will imply (11~); (11~“) Q(J) { JJJ} c { JJJ} c J T(J) implies (11~‘) by linearization, OE T(x)2y+2T(x) T((y)x= T(x)2y. Clearly if J is traceless then (11~‘) is trivially true.) We first form the upper layer J, of the sandwich: J is an ample outer ideal in the Clifford algebra J, = Qp,J for @i = @[e(J)] (which is a field extension of @ since by (IIv), D(J)* c @, so @c @, ~3). Indeed, J is outer in J1, U,,JcJ, by (IIiv) and (11~): for ai,/?,~@,, x, y~Jwe have U,,,J=ct~U,Jc@~Jc@JcJ and U,,,B,uJ=a,p,U,,,Jc~I{JJJ}cJ. J1 is Clifford over @I since the values assumed by Q on J, fall in @, by construction: &(J1) is spanned by &(@,x) = afQ(x) c @e(J) c cP1 and by

&A

PI y)=a,p,e(x,~)~~1~,~(J)c~1.

For the bottom layer of the sandwich, let Jo be the smallest ample outer @ideal in J (spanned by all U,, . . . U,” 1 for xi in J). By definition Jo 4 J is ample in J, and we need only show that it is Clifford over @,, = @[ T(J,)]. (Note that F(Jo)‘c &J)2c @ by (IIv), so @,j, is a field extension with @C@oC@ltJ-) @. The values assumed by Q on Jo all fall in @+,since &(J,) is spanned over 0 by all &Ux, ... Ux,l)=Q(xl)*~~~Q(x,,)*~@ (by (IIv)), and all &( U,, . . U,” 1, U,, . . . Uym1) = Q( Ujm . . . U,, U,, . . . U,, 1) E F(J,,) c @,, by definition (note J,, = I,,). It only U&l, l)=P(ujm... remains to prove that Jo is a @,-algebra, i.e., T(J,) J,, c Jo. It suffices to prove F(J,) 1 c Jo, thus to prove

n K, . . . U,” 1) 1 E Jo by induction on n. The result is trivial for n = 0 (T( 1) = 2 = 0). If true for n- 1 then F(Uu,, ... U,” 1) 1 = &( u,, Y, 1) 1 (Y = ux, . . . UX” 1) = &(A ui, 1) 1 = Q( y, x:, 1’ y ~x:+~(‘(y)x:+~(x:)y=y~x:+U,,{~(:(y)1}+~(x,)*yE

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MCCRIMMON

AND ZEL'MANOV

J, 0 Jo + U,{ J,} + @Jo (because F(y) 1 E Jo by induction, so j = F(y) 1 yeJo, and p(xi)* E @ by (11~)) cJO, completing the induction step. This finishes 6.5 and 6.1. 1 n 6.6. Remark. We cannot always assume that the lower layer of the sandwich is Clifford over @: if J= Qe, 0 Qu12@SZpe, for @ c ~2c fi, &uiz) = fl E Q but p $ @ = {c?)E s;i 1852 c Q}, then Jo = Qoe, @ Qu,~ @ Qoe, for Go = T(J,) = &(J,) = @[PI; here Jo is Clifford only over QO, not over @. There is an isotope J’“’ for u = e, + pe, which is strongly connected, @“‘(u,~) = 1, and then (J’“‘), = @e, @Quiz @ @pe, is Clifford over @ but not ample in J: it contains 1(“) = u-i but not 1. Thus the minimal ample outer ideal in J(“) is a function of u, and the field over which it is Clifford varies with u, and need not equal @ for u = 1. n

7. CLIFFORD IDENTITIES

Clifford identities are those which separate algebras of degree 62 from those of degree 23. In this section we show that the existence of a Clifford identity (no matter how complicated) on a nondegenerate algebra forces the standard Clifford identity [[x, y]‘, z, w] = 0 on that algebra. As a consequence, algebras with n 2 3 interconnected idempotents cannot satisfy any Clifford identity. Although we obtain our results for nondegenerate algebras, we will quickly reduce to (semi)primitive algebras; the advantage is that semiprimitivity is a “local” property: (7.1)

if z 0x, Uzx are nilpotent

for all x E J then z E Rad(J).

Recall that J satisfies a polynomial identity of the free Jordan algebra FJ(X) if

f = 0 for an element

f (x1, .... x,)

(7.2) (i) f vanishes strictly on J: all specializations offin all scalar extensions Jn vanish, f(a,, .... a,)=0 for all a,eJ,; (ii) the image f 0 off in FSJ(X) is manic: it has coefficient 1 of some leading monomial. Note in particular that f” #O, so we do not count s-identities as being polynomial identities. Also note that we make no monicity assumption about the Jordan expression for f in FJ(X), only for the image f” in FSJ(X). The standard process of linearization shows that we may always

167

STRONGLYPRIMEQUADRATICJORDANALGEBRAS

replace f by a multilinear identity, which (by relabeling the variables) may be assumed to have the form (7.2’)

f"(X1, '..,XJ =x1 . ..x.+ C a,x,(II ...xncnl 7tfl

for a, E @. We say J is a P.Z. algebra if it satislies a polynomial identity. We say J is of Clifford type if it satisfies a cliffard identity, a polynomial identity f = 0 as in (7.2) where f is a Clifford polynomial (does not vanish on the split matrix algebra H, = H(M,(@), t) of hermitian 3 x 3 matrices over @). Then f does not vanish on any J containing a copy of H, (hence any H, for n 2 3), and f does not vanish strictly on any J having a scalar extension s2J containing H,. We have not assumed that a Clifford polynomial does vanish on all Clifford algebras, so we include “degree 1” identities like

(i.e., {xyz} = { yxz}, i.e., [x, z, y]” = 0, in view of (0.27), (0.27’)). Rather generally, if certain hypotheses would force an identity in 63 variables on special finite-dimensional simple Jordan algebras of hermitian matrices, then they force the identity on any i-special semiprimitive algebra. The limit 3 arises because by Cohn’s results only for ~3 generators do we know that a special algebra looks like H,(A, *), and we need H,(A, *) to apply the powerful associative theory of prime PI. algebras. 7.3. GLOBAL IMPLICATION THEOREM. Let the set XJ(yl,...,yr) of hypotheseson elementsof a Jordan algebra be hereditaty in the sensethat it is inherited by subalgebras, scalar extensions, and homomorphic images: if %xY,, ...Ty,) holdsfor somey, , .... yr EJ then (Hi) the y,; (Hii) s2J~ J;

Ko(y,, %b(yl,

.... y,) holds for any subalgebra J, c J which contains .... y,) holds for the images yj in any scalar extension

(Hiii) S(j,, .... y,) holds for homomorphic image J of J.

the images ji

of the y, in any

We say a polynomial p(a,, .... a,) in elements a, E J vanishes locally if p(ci,, ..-a,,) = 0 for the images ciicrof ai in any local matrix algebra .i, (a local algebra of J is a scalar extension J, = 8a J, of a homomorphic image J, of a subalgebra Jo of J containing the ai, which is also a matrix algebra of the form 1, z H,(M,( c,)) f or a composition algebra C, over 8,). We say p vanishesglobally $p(a,, .... a,) = 0 in J itself for the given ai.

168 !f

MC CRIMMON

aa,

9 --.,

AND

ZEL’MANOV

a,) holds for elements a,, .... a, of a Jordan P.I. algebra J,

then (7.4) (r = 3) Nilpotence Principle: if J is i-special and p(a,, a*, a3)’ = 0 locally for some fixed n, then p(a,, a,, a3) is globally nilpotent; (7.5) (r = 2) Vanishing Principle: if J is semiprimitive i-special ~(a,, a*) = 0 locally, then ~(a,, u2) = 0 globally; (7.5’) (r = 1) Vanishing Principle: if J is semiprimitive then p( a, ) = 0 globally.

and

and p(a,) = 0 locally,

Proof. Say J satisfies a polynomial identity f =O as in (7.2). Fix an arbitrary element y E J and take Jo to be @(a,, y) if r = 1, to be @(a,, u2, y) if r = 2, and to be @(a,, u2, u3) if r = 3. Any nil-free J,/Nil(J,) E fl J, is a subdirect sum of prime nil-free images J, of Jo (cf. [9]). We now bring in the powerful associative P.I. theory to show the images J, are local matrix algebras. Each of these images J, of J,, c J is i-special since J is for r = 2,3 by hypothesis in (7.4), (7.5), and for r = 1 by the Shirshov-Cohn theorem [ 13, p. 7671 (in the quadratic case, any 2-generator algebra without extreme radical is special). Moreover, each of these J, is generated by < 3 elements since J,, is, so by Cohn’s theorem [ 13, p. 7671 these J, are special of the form H,(A,, *) for some *-tight associative envelope A, of J,. Here A, inherits *-primeness from J, by (0.19), and inherits an associative polynomial identity: H, inherits the manic f O(xl, .... x,)=0 from J by (7.2), and XEH(A,, *)*x2 = xx* E H,( , *) by ampleness (0.8’), so H(A,, *) satisfies f(x:, .... xf) = 0, hence by Amitsur’s theorem [25], A, itself satisfies a polynomial identity. But then by Posner and Rowen A, has a scalar closure which is linitedimensional *-simple, hence has a scalar extension 2, = $-,A, over an algebraically closed field 6, of the form (i) M,($-,) with transpose involution, (ii) Mln($=) with symplectic involution ( z M,,(Q($-,)) with conjugate transpose involution for a split quaternion algebra Q(5a), or (iii) M,($,)@M,(G,)“P with exchange involution ( z M,(G,EEi$zP)). In all cases A, = M,(C,) for a composition algebra C, over 6 with standard involution. Then the scalar closure J, = 8,J, of J, in 2, is an ample H,(& *), hence a finite-dimensional simple special Jordan algebra of the form H,(M,(C,)), so J, is a local matrix algebra. Since the hypotheses a,) hold in J, by (Hi) they hold in the subalgebra J,,, by (Hiii) in 8x4 9 .... its homomorphic images J,, and by (Hii) in their scalar extensions J,. If the hypotheses ZJ(a,, .... a,) imply q(a,, .... a,)=0 locally, then a,,) holding in each local J, forces q(Z,,, . . . . a,) =O, so JG&G, .... in each J,, q(aI, .... a,) vanishes in J,/Nil(J,,), thus 4(%r ***,a,,)=0

da 1, .... a,) E Nil( J,,).

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PRIME

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JORDAN

ALGEBRAS

169

For (7.4) with q=p” we see q(a,, a,, a3)eNil(J0) and therefore ~(a,, a2, u3) is nilpotent in Jo c J (but with no estimate of the index of nilpotency). For (7.5), (7.5’) with q =p we see ~(a,, .... a,) oNil(JO). But when r = 2 or 1 the element y is included in J,, (note this is possible only for r < 3), so also yap, UP y fall into the ideal Nil(J,) and hence are nilpotent in J. This holds irrespective of the choice of y E .I, so by (7.1) p lies in the radical of J. But we explicitly assumed J was semiprimitive in (7.5), (7.5’) Rad(J) = 0, so in these cases p = 0 vanishes as desired. 1 A Clifford identity f= 0 on J forces all local matrix algebras 3, of J to have degree n < 2 since they inherit the Clifford identity and thus cannot contain Z-Z(M,(@)). Thus J, E H,(M,(C,)) or H,(M,(C,)) for a composition algebra C,, hence all J, are Clifford algebras J(Q,, 1) relative to Q,(x) = det x E 6,. All of these .I3 satisfy the standard Clifford identity (7.6)

ccx,v12,wl=o

(7.6’)

[x,y]20z=o

(if2@=0)

since in fact all [x, y12 = {T(x)’ Q(y) + TV Q(x) - T(x) T(y) Q(x, y) 2Q(x)Qb)+QW,', 1 are scalars (note in characteristic 2 that [x, y] = xoy is a Jordan element, but in general only [x, y12 exists inside the Jordan algebra (cf. (0.25)). Roughly speaking, algebras satisfying Clifford identities behave like Clifford algebras (at least locally), so it is not surprising that the nice algebras satisfy the standard identity. 7.7. THEOREM. Any i-special satisfies a Clifford identity satisfies

nondegenerate Jordan algebra the standard Clifford identity.

which

Proof: It suffices to prove this for primitive algebras: by [8] any nondegenerate algebra J has an imbedding in a semiprimitive algebra 7 having exactly the same identities as J, and 3n n lE for primitive algebras Ja = J/KU which inherit all identities of 7. Thus in particular each JR is i-special and satisfies a Clifford identity; if we can prove each 3, satisfies the standard Clifford identity, so does the subalgebra JC fc n 1%. So let J be i-special primitive and satisfy a Clifford identity f =O. By primitivity J is an algebra over its centroid r, which is a field, so we may as well replace @ by r and assume @ is a field. (We make no further use of primitivity, only of semiprimitivity.) Fix a, b E J and let (7.8)

c= [a, b12

170

MCCRIMMON

AND ZEL'MANOV

as in (0.25). We wish to prove (7.9) (7.9’)

cc, 4 Yl = 0 cox=o

for all x, y E J for all XEJ

if

2@ = 0.

We have general principles 7.3 for deriving identities in <3 variables in i-special algebras, but this standard identity (7.9) involves four variables, so we must approach it step by step. Recall that all local matrix algebras of J are Clifford algebras. We first claim that for an infinite set T= {t,, t,, ....} of indeterminates (7.9a)

[c, x, x] is nilpotent

for each x E J[ T]

(and if 2@ = 0 then each COx is nilpotent). This follows from the Nilpotence Principle (7.4) applied to J[ T] with a1 =a, a,= 6, a3=x, X’=@, p(a,, a2, a,)= [[a,, a2J2, a3, a3] (resp. in case 2@=0 p(a,, a2, a,)= [a,, a2120a3): we saw that [a,, a,]‘E@l, so p=O locally in the Clifford algebras J,, hence p is nilpotent in J[T] by (7.4). This result depended on the internal structure of c. If we start from (7.9a), irrespective of how c came to satisfy that condition, we can by semiprimitivity apply the Vanishing Principle (7.5) to a, = c, a2 = x, ZJ= {[a,, 6, b] is nilpotent for each bEJ[T]}, p(a,, a,)= [a,, a2, a21 (resp. XJ= {all a, 0 b are nilpotent}, p(a,, a2) = a, 0 a2 if 2@ = 0). Note that XJ(al ) holds by our assumption (7.9a). To see that 2 is hereditary, subalgebra inheritance (Hi) is clear since J,,[T] cJ[T], image inheritance (Hiii) is clear since J[ T] = J[ T], while for extension inheritance (Hii) each b = C o,q, E QJ[ T] is a specialization of z = C tiqi for tj E T not appearing in the qis (here it is crucial that T be infinite, which guarantees that each Jn[ T] = JITln is “locally” a homomorphic image of J[ T], reducing scalar extensions to images; note that the hypothesis X’ that [a,, 6, b] be nilpotent for all bEJ need not be inherited by scalar extensions), and therefore nilpotence of [a,, z, z] implies nilpotence of [a,, b, b]. Thus %,(a,) is hereditary and holds in J, and by 5.1(I) it forces p(a,, a,)=0 locally in the Clifford algebras, so by (7.5), p = 0 globally in J. This is the three-variable version of (7.9) where x = y: (7.9b)

[c,x,x]=O

for all x E J

(and cox=O

if

2@=0).

If 2@ = 0 then c 0 x = 0 for all x by (7.9), and J satisfies the characteristic 2 version (7.9’) of the standard identity (which trivially implies the standard identity (4.9)), so assume @ is a field of characteristic 22 (hence

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ALGEBRAS

171

1~ @). For a change it is the linear Jordan case which is more complicated here. As with (7.9a), we can start from (7.9b) to get the result (7.9c)

all [c, x, y], [x, c, y] are nilpotent

for

x, ~IEJ.

We apply the Nilpotence Principle (7.4) to a, = c, a, = x, a3 = y, %$(a,)= {[a,,b,b]=O for all bEJ), p(a,,a,,a,)= [al,uz,a,] or [a,, a,, a,]. The hypothesis 2 is hereditary (scalar extensions pose no problem, since the hypothesis is quadratic in 6), holds in J by assumption (7.9b), and again forces p = 0 locally by 5.1 (I), so p is nilpotent in J by (7.4). A few passes with a magic wand, while one mutters the word “derivation,” removes the word “nilpotent” from the second half of (7.9~). By (7.9b), c belongs to the “alternative nucleus” Na,, = {de JI [d, a, a] = 0 for all a E J}, and the alternative nucleus is derivation-invariant ( [Dd, u, a] = D([d,u,u]) - [d,Du,u] - [d,u,Du] = D([d,u,u]) - [d,Du+u, Du+u]+[d,Du,Du]+[d,u,u]=O), so d=-D,,,(c)=[x,c,y] (by (0.28)) lies in N,,, yet is nilpotent by (7.9~). But N,,, contains no nilpotent elements, (7.10)

if z is nilpotent

and [z, x, x] = 0 for all x E J, then z = 0,

since this is true locally in Clifford algebras by 5.1(11), and we can apply the Vanishing Principle (7.5’) to a, = z, XJ = {a: = 0, [a,, 6, b] = 0 for all bEJ), p(u,)=u, (note ;ri”J(ul) holds by hypothesis (7.10) and again is hereditary), yielding that p vanishes globally and thus that a, = z is zero in J. Applying (7.10) to z = d = [x, c, y] shows (7.9d)

c&c,yl=o

for all x, y E J.

Since f E @, the middle nucleus and center coincide [23], so [c, x, y] = 0 too, and we have established (7.9). 1 Recall that n orthogonal idempotents ei (ey = ei, eio ej = U,ej = 0 for i # j) are interconnected if YJ(ei) = J for each i (which forces eje U,$Jii if n > 2, and Jij= J,o Jki if n > 3, for distinct i, j, k), where the Peirce spaces are defined by Jii = U, J, J, = U,, e,J. We say J is doubly interconnected if it has n > 3 idempotents with $5(J,20 J13) = J, or n > 2 idempotents with yJ( [ J,, , J12, JI1]) = J. Note that such e, , e2, e3 are necessarily interconnected: XJ(ei) 1 $(J,, 0J23) and 9J(ei) 2 Sl,( [J,, , J,,, Jll]). Conversely, all algebras with n 2 3 (but onfy certain ones with n = 2) interconnected idempotents are doubly interconnected.

172

MCCRIMMON

AND ZEL'MANOV

A Jordan algebra Jcannot satisfy the standard cltfford identity if it contains three orthogonal idempotents whose Peirce spaces have J,,Q Jz3#O, or if it contains two orthogonal idempotents with [J,,, J12, J,,] (= [[JII, J,,], J,,] = [VJ,,, VJ,,] J,2) # 0. Moreover, tf such an algebra is nondegenerate and i-special, then it cannot satisfy any cl&ford identity whatsoever. An arbitrary i-special algebra cannot satisfy any Clifford identity whatsoever zf it contains three orthogonal idempotents with 0 fe, EY~(J,,~ J,,), or rf it contains two orthogonal idempotents with 0 #e, EYJ( [Jll, J12, JII]). In particular, a doubly interconnected special algebra cannot satisfy any Clifford identity.

7.11. CLIFFORD INTERCONNECTION

THEOREM.

Proof: For x,EJ~ we have [e,+e,,x,,+.x,,]2=[e,X]2=xoU,XU,X* - U,e* (Cf. (0.25)) = 0 - U,(Xf, + Xl*” X23 + X:,) -EII(X~~)-X~~~~~~

-E33(XZ3)-E22(UX,2e~

+ Ux2,e3)

Vu

U,(e,

+ e3) the

=

Peirce

projections) so [[e, x12, e,, e3] = ([e, x]*oe1)oe3 = -x120x23, therefore the standard Clifford identity would force J120 J23 = 0. Similarly [xii, y,,] 0 Yll” ux,, P, Y,*+YL*“Ux,, c, Y11x,,,.,hoh2)Uv,,.Y,2(xIl~eJ ==O+~,~“(x~~‘“~~,)-O-Z(y,;x,,ylz) = ccxH>Y**l,Y121 = = {x11 Y,, Y,21- {Y llXllY121 = [K/,,,? VJY12 -CXI~,Y~~,Y~~I, so CCX~I,YIII, ~~~l,e~,e~l= IICX~~,Y~~I,Y~~I, therefore the standard Clifford identity would force [ [Ji,, JI1], J12] = 0. If J is nondegenerate and i-special, then any Clifford identity would force the standard one by 7.7, and if J is arbitrary i-special then a Clifford Eiel)y121+Cel,y,,l”CX,,,y,,l

identity

would force the standard one on the nondegenerate quotient where if O#~,E~~;(J,~OJ~~) or YJ([[JIL, J1,], J,,]) then 0 #C, E&(J,, oj,,) or &([I [I,,, J,,], j,,]), contrary to what we just algebras 1. (Note that e, #O*ei not proved for nondegenerate nilpotent =c-e, $9(J) c Nil(J) * e, # 0 in 1.) 1

J=J/Y(J),

8. CENTRALITY

OF [x,y]*

We have seen that any i-special nondegenerate algebra satisfying a clifford identity satisfies the standard identity [[x, y]*, z, w] = 0. In this section we start from any prime nondegenerate algebra (not necessarily i-special) satisfying a standard clifford identity, and establish the centrality of [x, y]‘. This is immediate for linear Jordan algebras. In the quadratic case the notion of center is much more nebulous; we prove such J has a unital scalar extension j= 03 over an algebraically closed field 6 which satisfies [x, y]* E 6. This requires some preliminary results on centers and centroids. To view J as an algebra over Q, the elements of 52 must act as linear operators (scalars) on J. In certain cases we can identify these operators

STRONGLY

PRIME

QUADRATIC

JORDAN

173

ALGEBRAS

with elements of J. Any linear algebra may be considered as an algebra over its center (8.1)

[ c, x, y] = [x, c, y] = [x, y, c] = 0 for

C(A)= {c~Alcx=xc, all x,y~A}.

In the case of linear Jordan algebras the center reduces by commutativity to (8.2)

C(J)={c~Jl[c,x,y]=O

for all x, y E J}

C(J) = T(J) 1

if

SE@);

J is unital.

For quadratic Jordan algebras this definition in unsatisfactory, because we cannot make J into a C(J)-module in a natural way. (If J = Q + is commutative of characteristic 2 then C(J) = J since co x = 0 for all c, x -the circle product c 0 x z cx + xc does not accurately capture the action c .x = cx = xc, which indeed cannot be expressed in Jordan terms.) Rather than get sidetracked on the general question of central elements, we will deal exclusively with scalar elements, where an element c of a unital Jordan algebra J over @ will be called scalar if CE@l.

(8.3)

We need a criterion that will guarantee V,, U, are centroidal. 8.4. LEMMA. An element c E J will have V,, U, E f(J) they are in the outer-centroid, for all

c~,~~.Yl=cvc~~.~l=o

(i)

in the centroid iff

xE.?.

Centrality of V, alone already goes a long way towards forcing U, : we have V,, U, E r(J) iff

(ii)

Vc E r(J),

so in characteristic

(iii)

centrality

of

Uucc,s. i = 2U.x U,,

2 it suffices if

vc = 09

U”,,. ,x,zi= 0

(2@ = 0).

In particular,

(iv)

Certainly centroidness

Proof:

inner

V,=O

*

vc2 ( = 01,

UC2E r(J)

(i) is necessary for the centroid

(24,=0).

(4.2), and it implies

Uucr) I = U,U,U, (by (0.3))= Uf U, and U,,,,= ?+Yx-2U,U, UC ux + ux UC + vc U.Y vc - U”(E)& .Y (by (0.20)‘)=2U,U,+ (note { U,x y x} = V.x,.vU,x = U,. V.Y,Yx = U,(2U, y)) = c U,. To see that

174

MCCRIMMON

AND ZEL'MANOV

(ii) suflices, note that it implies (i) since [UC, U,] = U,U, + u, u,. - 2u.r u,. = ( UC.>i + u”(+. r - V,. U, V,} - 2U, U,. (by (0.20)‘) = U”(4 1. .x- 2U,U,. if V, E f(J). Part (iii) is a special case of (ii). For (iv), note that I’,. = O+ V,.Z= e =0 (by (0.23)), and by (0.33), UUCr+s = - vc2” x vc2,I + 2u,2 ux + vc2 u, v,z + v-,2,cx. c)2 - vc2, c ,;UCx)(’ - v,3, c ; $ + 1 vc4Yx. x =0 so by (iii), V,,Z, UC2Ef(J). Note that if V,, U, E T(J) for J of characteristic 2, then V, must vanish if J is semiprime: it lies in the centroid and is nilpotent, V,‘= V/, VC2 (by (0.23)) = V,,.z + UC,,2 (by (0.22)) = Vv~C~C + U, I’,. (by (0.24), (0.2)) = U, V,. + U, VC= 0 in characteristic 2. 8.5. Remark. It would be interesting to know whether V, E f(J) implies U, E T(J) for nondegenerate J (even if only in the case V, = 0, 2@= 0). In the latter case one can show Z, x = [U,., U,] = UvCCjX, x = VuCox,~ +

N {c, x, c, x, } is a nilpotent (2ztj;i;)2

= 0.

derivation

with nil range, Zf, x = 0 and

n

We also need a condition

guaranteeing

scalarity.

8.4. LEMMA. If J is a unital nondegenerate Jordan algebra of characteristic 2, whosecentroid @ is square-root closed, then any element c E J with coJ=O is a scalar c~@l. Proof From V, = 0 we deduce U,ZE I’(J) = CDby Lemma 8.4(iv), hence c4 = U,,(l) = yl E ~$1. Since CDis root-closed by hypothesis, we can write y = A4, c4= A41, so in characteristic 2, z= c+ll has z4=0 as well as z 0J= 0. But we have the general principle [S, Lemma 2, p. 221

(8.7)

If J is nondegenerate then

z2=zoJ=0

=z. z = 0.

(Indeed, by nondegeneracy z2 = 0 * z3 = 0 (note U,, = U,ZU, by (0.3)), and by linearized (0.3) when zo y = z2 =z3 =0 we have 0 = U,{ U,,, + uz2, y2 u: uy - uy u: - u: y > = - uz2 uy - U”(z) y - { vz2 vz, y - vz3* y > u, y (by (0.3), (0.24), (0.22)) = - Uu(z),v Then nondegeneracy gives U, y = 0. If this holds for all y then again by nondegeneracy z = 0.) Thus in our case (z2)’ =z4=0 and z ‘oJ=zo (zoJ) =0 (by (0.23)) yields z2 =0 by (8.7), then z2 = zoJ= 0 yields z =0 by (8.7) again. Once z =0 we have c=AlE@l. 1 Now we are ready for the main scalarity theorem. 8.8. THEOREM. If J is a nondegenerateprime Jordan algebra satisfying a standard &ford identity, then there is a tight unital-scalar extension 3 = 03 which is prime and nondegenerateover an algebraically closedfield a such that [x, y]‘~fil for all x, yeJ.

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ALGEBRAS

175

ProoJ Since the standard Afford identities (7.6), (7.6‘) are quadratic, and all their linearizations vanish when any of the variables is replaced by 1, any unital-scalar extension J= df will continue to satisfy the standard Clifford identity. By 4.3 there is a tight unital extension 7 which is unital prime nondegenerate with centroid d an algebraically closed field. If the characteristic 22 then (7.6) shows [x, y]* E C(3) = r(s) 1 = s’il by (8.2). If i characteristic=2 then (7.6’) and (8.6) show [x, y]‘~fil.

9. ALGEBRAS

WITH [x,y]*=O

One trivial way for an algebra to have all [x, y]* scalars is to have them all vanish. In finite-dimensional simple algebras this identity forces [x, y] = 0, i.e., J is of degree 1. We show that this holds in general. There are complications in characteristic 2 since a degree 1 algebra need only look like J = Q,, 1 for an ample Q, c Q, not like all of Q+. Here primeness is not important: nondegeneracy suffices. We can completely JC Q2t by a polynomial

characterize identity.

the nondegenerate

Jordan

algebras

9.1. DEGREE 1 THEOREM (cf. [32]). The nondegenerateJordan algebras J satisfying [x, y]’ s 0 are precisely those J which can be imbeddedin a commutative associative @-algebra Q having no nilpotent elements, JC Q+. In particular, they are all special. Zf J is unital and central (T(J) = CD)over an algebraically closedfield CD,then J = @1. Proof Because of nondegeneracy we know the centroid (4.2) is a commutative associative @-algebra, containing no nilpotent elements (T* = 0 implies U T(Jj = 0, hence T(J) = 0 by nondegeneracy, so T= 0), and J is a r-algebra. We begin by proving the absence of nilpotent elements in J,

zn=o

(9.2)

*

z = 0.

For this it s&ices to prove z* = 0 =z =O. So suppose z2 =O. Then the hypothesis [x, y]’ = 0 means by (0.25) ydJ,y=u,y*+u,,x*;

(1) when y=z (2)

we see

z,=u,z = u:x*

(z' = O),

hence O=U,{U,~,(.~,;-U,,,,*2}=UI{CUZUU(.~)r+U”(x,:Uz+Ut.U(x)z - Ur~,CUCrI r,~] - U=U,Z U,} (by linearized (0.3)) = 0 + Uz UuClji UT+ 0 - 0

176

MC CRIMMON

AND

ZEL’MANOV

=U U(z) U(l) i (by U;U,= U,Z=O and UzUz,U(x,r= U,Z,,,~V,,,U,? z u, - uz2. U(x)22- Uz? x uz, x + Uz4,x~=0 by (0.33) when z2=0, noting that z4 = (z’)* = 0 and z3 = 0 by nondegeneracy), so by nondegeneracy

uz u,z = 0.

(3) Linearizing

x + x, y in this gives U, { x z y } = 0, so by (0.2)

(4)

u:, U(z).P = 0.

Since the algebra J’ = J[t] = JO, @[t] of polynomials in a scalar indeterminate t remains nondegenerate and inherits the quadratic identity [x,y]‘=O, we can apply (3) to z’=z+ tU,y (note z” =z2+ tUz2,ry+ t2UZUvz2 = 0) and identify coefficients of t to see that 0= U,U,(U, y) + U z,U(z)y uxz = UU(z) x Y + 0 (by (0.3), (4)), so Uuczjx J= 0 forces U,x = 0 by nondegeneracy, and Uz J= 0 likewise forces z = 0. Thus (9.2) is established. In the absence of nilpotent elements we show that the inner derivations collapse, D,,,(J) = [[x, i], J] = 0, so by (0.27) x~(y~w)=y~(x~w).

(5)

It suffices to show that D,,,(w) is nilpotent, where by (0.30), D,.Jw)~ easy to see that w ‘I UCx.yl w - UCX.Yl w* (since [x, y]’ = 0). It is particularly this vanishes for i-special algebras (the ones we are really interested in): these satisfy Thedy’s s-identity (0.32), so ( Ucx,),, z)’ = Ucx,Y, Uz[x, y]* = 0 shows U,,, Y, z = 0 for all z, in particular for w and w2. For general algebras we cannot use (0.32) for general z, but we can use it for z=xn by Shirshov’s principle (0.33), therefore ( Ucx,v, x”)’ = 0 * Ucx,Y3 x” = 0 = D,,,(x)~ = 0 (using (0.30)) =z-D,,,(x) = 0, hence {x y x} = { y x x}, i.e., y~x*=2U,y=xo(xoy)-x*oy (by (0.22), (0.23)): 2u, =

(6)

vx2,

v2*=2v,*.

(7)

Then from (0.26), Ucx,y,=Ux~y-2U,~Uy=U~.y-V~2~Uy (by (6))= U *“Y - ux2-y,y (by (0.21)), so by (0.30), D,,,(w)~ = wo Ucx,u,w~CX*YlW * = (x*~Y~Y)

wo { u,,,-

ux2.y,y}

w-

{U,,,-

ux2”y,y}

w2

=

u,{(x~y)2

(l))= U,(([x, y]*+2U,yz+2UYx’)2x2 0y’} (by (0.26), (7)) = U,{O + x2 0y* + y2 0 x2 - 2x*, y’} (by (6)) = 0. Thus D,,(w)~ = 0 in all cases, and (5) holds. If f E @, (5), (8.2), (9.2) show that J= C(J) = IR + is itself a commutative associative @-algebra without nilpotent elements. If J is unital and central, then J= C(J) = T(J) 1 = @l. (by

1inearized

STRONGLY

PRIME

QUADRATIC

JORDAN

ALGEBRAS

177

Next assume that 25= 0. Then [x, y] = x oy exists inside J, and (x 0y)’ = [x, y12 = 0 forces x 0y = 0 by absence of nilpotent elements, (8)

JoJ=O

(2J= 0).

From (1) we see (9)

u, y2 = &x2

(2J= 0)

and hence {xYw}~= UXU,w2+ U,UYx2+ Vx,,,Uw,(xoy)- U,Uvwow (by (0.20))= WJ,Y~+ WJ,Y~+O-(by (9), @))= (L,+ Uu(r)w,wWLWY~ (by (0.2OY)=O+ uuwvv~~-O (by @)I= u.v(u,w~w) (by linearized (9))=0 (by (8)), so {xyw}=O:

(10)

{JJJ}=~.

Then for x,y~j we have [U,, UY]=UxU,+ U,U, (by 2J=O)= u,,, + U$,$ - u; y (by linearized (0.3)) = 0 (by (8), (lo), so the multiplication algebra ‘M(J) generated by all U, is commutative. In particular, all U, lie in f(J). We claim the @-quadratic mapping x -+ U, is a E-linear map J -+ r(J): it is additive since U, +Y = U, + U, + U,, y, where U,, y = 0 by (lo), and it is multiplicative since UufxlY = U,U,U, by (0.3). Thus J is imbedded in the Z-algebra T+ for r= T(J) (again note that U, = 0 => z3 = U,z = 0 *z= 0 by (9.2)). We can give T(J) a new, twisted @-algebra structure Q by

since a + a2 is an endomorphism of CDwhen 2@ = 0. Now the map J + Sz+ is o-linear, U,, = a2U, = a . U,, so again Jc Q+. If J is unital and T(J) = @ algebraically closed, then J= @l since any x has U, = A21 for IE@, z=Al+x has U,=12Z+Ux (by (lO))=O, so z=O by nondegeneracy, and x = Al. We split the case of general @ into a characteristic 2 part and a 2-torsion-free part, and apply the above to each part separately. In our case K, = {ZE JI2z=O} is an ideal (always 2’= {zIsome 2”z=O} is an ideal, but here this reduces to K, since 42 = 0 =E.(2~)~ = U,(8z) = 0 =. 22 = 0 by (9.2)). Similarly K2= {zEJIU,K,=O} is an ideal: it is outer and inner in the sense of (0.13) since by (0.3), U uw,K, =_UJJJJoK~~ U,(U,K,)=O and U uCrJ.K1 = Uz U,( U,K,) = 0 for any a E J, and it is a linear space since by (0.20) if zlrz2eK2, {z~~,z,}~=U,,U~,Z~+U~~U~,Z~+ V r,,k,Uz&oz,) U:,“k,Z20~2E U,,K,+U,,K,+V,,,U,,K,-V,U,,K, = 0 (since K, 4 J) forces U=,, zzK, = 0. Moreover, we have K, n K2 = 0

178

MCCRIMMON

AND ZEL'MANOV

sincezEK,nK2~z3=U,zEU,,K,=O~z=Oby(9.2).Thuswehavethe subdirect sum decomposition J c JJRl q J/K,.

Now ,I, = J/K, has no 2-torsion (22~ K, *4z=O* 2z=O) and no nilpotent elements (z”~K,=s2z”=O~(2z)“=O~2z=O~z~K, by (9.2)), so its 2-localization J, [t] has no nilpotents over @[t] containing i, so by the previous cases J, c J, [j] c Sz: where ,52, has no nilpotents. On the other hand, J/K, has characteristic 2 since 2Jc K, ( UzJK, = U,(4K, ) = 0), and has no nilpotent elements ( Uz2K, = 0 * ( U,, z2)2 = U,,U,~1(:=0~U~,z2=0(i)y(9.2))~(U,K,)*=U,U~,z2=0~U~K1=0

(by (9.2))), so by the previous case we again have J/K, cSZ:. Thus Jc (sZ,EElL22)+ where Sz =1;2,EEX?, is a commutative associative @-algebra without nilpotent elements. Conversely, any Jordan subalgebra Jc 52+ will have no nilpotent elements (hence will be nondegenerate) if Sz has none, and will always have

Lw12=Q

I

9.3. Remark. We can use our general principle 7.3 to get [J, J, J] = 0 (and JoJ= {JJJ} =0 in characteristic 2) in the case of special interest where J is semisimple and i-special. Indeed, heref(x, y) = [x, y12 is itself a Clifford polynomial, and if f= 0 in a local image 7, c J( Q, 1) then Ta (1) has no nilpotent elements, (2) satisfies [Ja,, JE,, Ja] = 0, (3) satisfies --&o&=0= (JsIJoLJa)=O in characteristic 2. These local properties pass to J: for (1) use (7.5’) on X= {a~=O},p,(a,)=a, to see z2=O*z=0 in J; for (2) use (7.4) on S= @, ~*(a~, a2, a3) = [a,, a2, a31 to see [x, y, z] is nilpotent in J, hence zero by (1); for (3) use (7.5) on X=0, to see x oy - 0 in J, and use (7.4) on 2 = 0, P*k,t a2)=al”a2 ~~(a,, a,, aj) = {ala2a3} to see {xyz} is nilpotent in J and hence zero by (1). I

10. ALGEBRAS WITH [x,yj'~@1

Here we show that if an algebra over a field @ has [x, y]* always a scalar but not always zero, then the algebra is the Jordan algebra of a quadratic form over @. Once more the linear Jordan case is easy, and the quadratic Jordan case quite messy-it requires algebraic closure of the field and nondegeneracy of the algebra. The fact that algebras with scalar [x, y12 are of degree 2 follows from the Hall-Zel’manov identity

179

STRONGLYPRIMEQUADRATICJORDANALGEBRAS

(10.1)

~c~,y12~x2-([x,x~y]~[x,y])~x

+

w%x~y12-

[x,y]~[x,.x20y])=o

and its weaker quadratic version (10.2)

u~~~c~~y12~+u,~c~,yl~c~,

~,yl-c~,~~y12~+cx,u,y12=o.

Since these are Jordan identities in two variables, it suffices by Shirshov’s principle (0.33) to verify them in associative algebras, where everything follows by direct calculation (note Lx, .uoyl =xCx,vl+ CX,YIX, [x, x2 oy] = x*[x, y] + [x, y] x2). These are just Lx, v,yl=xc.%Yl x, versions of the Hall identity c~,Y1~~2-~(c~,yl~~~,Y~l~~+c~,Y~~12=~,

which shows that alternative

algebras with scalar [x, y12 are degree 2.

10.3. THEOREM. Zf J is a unital Jordan algebra over a field @ such that [x, y]’ E @l for all x, y, but not [x, y12 = 0 for all x, y, then when char(@) # 2, J is generically algebraic of degree2 over @, x2-T(x)x+Q(x)

1 =O,

and J= J(Q, 1) is the Jordan algebra of the quadratic form Q with basepoint 1. The sameholds when char( CD)= 2 if @ is algebraically closedand J is nondegenerate. In particular, all such algebras are special. Proof: First we handle the easy characteristic # 2 case. The quadratic condition [x, y]’ E @l continues to hold in any scalar extension, so we may pass to an infinite extension and make use of the Zariski topology. If all [x, y]‘= q(x; y) 1 lie in 01 then also [x, y] 0 [x, z] = q(x; z, y) 1 E ~81, so (10.1) becomes

(10.1’)

4q(x; Y) x2 - 2q(x; y, XOY) x + (2q(x;x~y)-q(x;y,x2~y))

l=O.

From~EQiandq~Oweseeby(10.1’)thatx2~~x+~1foradensesetof x’s, hence for all x’s, so J has generic degree 2, and so, as is well known, has the form J(Q, 1). In characteristic 2 the argument is more complicated. The identity (10.2) becomes (10.2’) 607!69/2-4

4b;Yb4+Mx;Y,

U,y)-q(x;x~Y))x2+q(x;

U,y)

l=O.

180

MC CRIMMON

AND

ZEL’MANOV

From q & 0 we know J is generically algebraic of degree ~4: (10.4)

x4+clxZ+B1

=o

for H, /i E 0 (uniquely determined if x2 $ @l, and if x2 = Al E @l we agree that a = 0, p = A’). We wish to show that x satisfies an equation of degree 2. Choosing a root AE @ of A2 + al +p =0 (by algebraic closure of @) it suffices to show xi =x + Al satisfies a degree 2 equation, where now x;‘+axf

= (x4+A4)+a(x2+A2)

= (x4+ax2+/?l)+(14+aA2+/3)1=0.

Either a = 0, and XT = 0, or a # 0, and (by algebraic closure) x2 = a -‘/‘x1 has x4=ap2x4=ap1x2=x2 2, so we can normalize (10.4) to the two forms 1 2 1 (10.4a)

x4=0

(10.4b)

x4 = x2.

For (10.4a) we have a general principle (10.5)

x nilpotent *x2 = 0

using semisimplicity and the Vanishing Principle (7.5’) for x = a,, ~={a~=O},p(a,)= a:, since all local images Ja inherit the standard clifford identity from J, hence are Clifford algebras where a; = 0 implies LZ:= 0. Thus in (10.4a), x4 = 0 3 x2 = 0, and x trivially satisfies a degree 2 equation. In (10.4b), x2 = e is an idempotent; if e = 1 or 0 then x2 E 01, and again x satisfies a degree 2 equation, so suppose e # I,0 is proper. for xii E J, in the Peirce decomposition If x=x,,+x,~+x~ J=J,,@J,O@Joo relative to e,=e, e,=l-e, then x,,=eox-2x,,= characteristic 2, so x=x,1+x00, X2 ox-2xl, = 2x3 -2x,, = 0 in e=x2=x:i+x&, so x&=(e+x,,)2=0. But

(10.6)

e#l,O

*

Jll,Joo

contain no nilpotent

elements

since the Jii inherit nondegeneracy from J, and [Jii, Jii]’ c Jii n @1 = 0 Thus (since e# l,O), so by (9.2), Jii contains no nilpotents. and x satisfies a degree 2 equation in both the xoo=e+x ,1 --0 , x=e=x2, cases (10.4a) and (10.4b). Thus all x are degree 2. [ 10.7. Remark. For semisimple i-special algebras an alternate argument considers the two cases [x2, y12 = 0 and [x2, y12 & 0 separately in characteristic 2. If (x2 0~)’ s 0 then by the Vanishing Principle (7.5), x2 oy = 0 (since in local cases this forces degree 2, where we apply S.l(II)--note that we need i-speciality and semisimplicity to apply (7.5)), so linearization of the generic relation (10.4’)

x4+S(x)x2+R(x)1=0

181

STRONGLYPRIMEQUADRATICJORDANALGEBRAS

gives O=~~~(~~y)+S(x)(x~y)+S(x,y)x~+~,R(x) (*I

1, hence

S(x) xoy + S(x, y) x2 E @l.

If SrO then x4e@1, x4= J41, x= Al + w for w4=0, so by (10.5), w2 =0 and x2 = A21 E @l. So assume S f 0. If S(x, y) & 0 then (*) implies x2 E 01 + XOJ for a dense set of x’s (hence for all x), so replacing x by x2 we see x4~@1+x20J=@1 (recall x’oy=O), and again ~‘~01. If S(x, y) = 0 then (*) implies xny =f(x, y) 1 E @l since S & 0; here f(x,y) f 0 since (xoy)‘= [x,y12 & 0, so from f(x,y)x2= &.(xoy)= x0 U, y=f(x, U, y)l E @l (by (0.2)) we again get x2~ @l. Thus [x2, y]‘=O implies x’~@l for allx. On the other hand, if [x2, y12 # 0 we claim z = x2 + &x + a 1 vanishes (a = S(x), /? = R(x) as in (10.4), (10.4’)); by semisimplicity it suffices by (7.1) if all z 0y, Uz y are nilpotent for the Zariski-dense set of (x, y) where 4(x2, y) # 0. It suffices if the image z,, of z in J,= @[x, y]/Nil(@[x, y]) is zero; here Jo s n Ji for local Ji c J(Qi, 1i) over fields Qi 3 @, so it suffices if each component zi of z0 vanishes. But Jj is Clifford, x: = clixi + pi 1, for LX~,/?~E@~, x~=a:x~+/?~l,, so by (10.4), {a+af) x2= (b+/?2} li; here x: $ @l i (since 0 #4(x2, y) 1i = [xf, yi12), so by independence a + a: = p+j:=O, &=air &=fii, zi=x~+aixi+aili=O, as desired. 1

11. PRIME ALGEBRAS OF CLIFFORD TYPE Putting our results together, we show that nondegenerate prime i-special algebras of Clifford type are forms of Clifford algebras J( Q, 1) of nondegenerate quadratic forms, in particular they are special. Our results so far show that prime algebras of Clifford type over algebraically closed lields are Clifford algebras, and this shows by scalar extension that they all are. 11.1. CLIFFORD STRUCTURE THEOREM.

An i-special nondegenerateprime Jordan algebra of Clifford type (satisfying a Clifford identity) is a Clifford form: there is a unital-scalar extension J= fij of J having the form I= J(&, 1) for a nondegenerate quadratic form Q over the field 6. In particular, all such J are special. ProojI By (7.7) and nondegenerate i-speciality, J satisfies a standard Clifford identity, and by (8.8) there is f = fij which is i-special nondegenerate prime satisfying [x, y12 E s”il over the algebraically closed field d. If [x, y12 = 0 we have J”= 6’ of degree 1 by (9.1), while if [x, y12 f 0

182

MCCRIMMON

AND ZEL'MANOV

we have J= J(o, 1) of degree 2 by (10.3) (where e is nondegenerate nondegeneracy of 3). We can subsume degree 1 under degree 2 since a+ =.I@,

1)

for

by

Q(G) = I%’

with 0 nondegenerate (even anisotropic) when fi has no nilpotent elements. Thus in all cases J= J( &, 1 ), which is well known to be special with universal envelope the Clifford algebra C(&, 1) of the quadratic form e with basepoint 1 [S]. 1 Note that speciality of algebras of Clifford type comes out only as an afterthought at the end, not as an essential tool from the beginning as with algebras of hermitian type. If we could establish speciality earlier we could avoid much of the mess in Section 8 concerning centers in characteristic 2: from c 0 J = 0 in characteristic 2 we would get c E C(A) for any envelope A for J, hence c would be a scalar in the extension C(A) J. PART

III.

CONSTRUCTION

OF ZEL’MANOV

POLYNOMIALS

Having seen the strong structural conclusions that flow from the existence of a hermitian part or a Clifford identity in a prime algebra, we now construct specific examples of Zel’manov polynomials. Although the exact nature of such a polynomial is largely irrelevant, we will try to construct as simple an example as possible, and show how such polynomials sit in the free special algebra. All our calculations take place in the free special Jordan algebra inside the free associative algebra.

12. EATERS It is technically awkward to construct an ideal z(X) a f(X) just from the multilinear hermitianity condition (1.2). We saw in 1.6 that {J...J~‘“‘}c{~...~}c~, so Z(n) will have to eat n-tads. We will concentrate on this property of eating n-tads. In linear Jordan algebras the tetrad-eating ideal is hermitian and consists precisely of the pentad-eating elements. We say that an element p(x,, .... x,) E%(X) of the free special Jordan algebra j(X) on an infinite set of generators ears n-tads if whenever we put it into an n-tad in last place it chews up the n-tad and spits out a Jordan polynomial. Universally this means (12.1)

{Y,...Yn-IP(X*,...,Xm)}=P’(X,,...,Xm,Y,,...,Y”-,)~f(~uY).

STRONGLYPRIMEQUADRATIC

183

JORDAN ALGEBRAS

We denote by

E, = E,(X) the linear space of all n-tad-eaters in f(X). E,xE51

Here E, = E2 = E, = y(X)

>

. . ..

We say an ideal Y(X) Q f(X) eats n-tads if all its elements do. The advantage of the linear condition (12.1) in contrast to (1.2) is that it is preserved by sums, so there is a unique maximal n-tad-eating ideal

namely the core of E,,: (12.2) TnW)=Elj=

‘&XkW-1

I {~r-~,-d&))

ccY(Xu

0

Once more we have T, = T2 = T3 = f(X) > T4 1 T, 2 .. .. With yn-, E x a # we see by homogeneity that the Jordan polynomial p’ in 9.1 has a factor in &” and hence fall back in xx, (12.3)

{Y, --Yn-20)

CL%

if

% 42,

PET,(X).

In particular, { T,, . . . T,, T,,} c T, is n-tad closed. For n = 4 we see that T4 is tetrad-closed, so in the linear case is hermitian by (1.2,,,): (12.4)

T4 is hermitian

if f E @.

Note that E, and T,, are invariant under all linearizations, and under all endomorphisms and derivations of $ (since these extend to the special universal envelope a, hence can be applied to the associative-not-Jordan products { y i . ..y.,} in (12.1), (12.2), where by disjointness the substitutions in X leave Y alone). Because there is no finite formula for y2(p)=C;Z0 U>(p+ U,f), condition (12.2) for n-tad-eating is hard to relate to p. It turns out that it is pentads (rather than the tetrads one would have expected from Cohn’s theorem) that are crucial as far as eating goes. 12.5. F'ENTAD-EATER THEOREM. The set E,(X) of all pentad-eater forms an outer ideal in B(X), and its core 6)

TAX)=

(PEE* I u,Yc&j=

IPEG

IP’E&)

is the pentad-eating ideal. This is a linearization-invariant T-ideal. We have

(ii) (iii)

W(X)

= T,(X) = TdX)

27-O’) + Tc,(W’=

E,(X)

184

MC CRIMMON

AND

ZEL’MANOV

so for linear Jordan algebras (iv)

T4u-) = T,(X) = E,(X)

(fE@)

and the hermitian tetrad-eating ideal consists precisely of all pentad-eaters. Z(X) is any hermitian ideal, then &yx)‘3’

(VI

If

c T&r).

Proof The equivalence UP j c E, o p2 E E6 of the two formulations (i) for p E E, follows via the identity

in

= {Y,,Yz,({Y,,Y,,POY,)-{Y,,Y,,Y,,P}),PJ -

hYZd5~

{Pdd3}9!4+

{Yld2d5d3dhP2)*

Outerness p E E, S- U, p E E, follows via the identity

= iYl> (Y,,Y3,Y4,Y1PltY)-YloI(Y3Y2Y),Yq,Y,P1 + {Yl, {Y3Y,Y},Y,,Y,P}+Y,o{Y,,Y*,

- {Yl?

Y3,

Y*,

qy.4,

U,Y4,Pl

PI.

Whenever K is an outer ideal in J, the core of K is explicitly K”=

{zEKI

given by

t&kK}.

(Clearly Ic K, Id J=s iJ,jc Zc K+ I c p, and conversely p is already an ideal: it is a linear space since 17,,,~,j = V,,,jz, c V,jK t K by outerness of K, it is outer U,ZG Ko as in (0.13) since U,,tc UjKc K has Uu(,,,,j= U,, U,U,,j (by (0.3)) c U,( U,j) c U, KC K by outerness of K, and it is inner U, ye k? as in (0.13) since U, ye U,,?C K (by definition of A?) has U uCzjvj= iJ,U,,U,j (by (0.3))~ U,Jc K.) Thus g= T5 a f as in (i). In (ii) we know that T4 3 T,, and we have T, 2 2E, since for p E E, we haveA 2U,j= {p fp} = V,,jp c ES by outerness (i), hence U,j= 4U,,Jc E5 shows by (i) that 2p~ T,. For (iii) we show p E T4 = 2p, pz E E,. For the first note that p E T4 => all U,pe T4c E,=z-all {y,, y3, y,, y2,p} E$ by the identity

STRONGLY

PRIME

QUADRATIC

JORDAN

ALGEBRAS

185

(set y, = 1 in (12.7)), so linearizing y + y, 1 yields 2{ y,, y3, y,, y, p> E f and 2p E E,. For the second p2 E ES note that p E T4 * all U, y E T4 c E, * p2 E E, by the identity (12.6’)

UpY4)

{Y,kY2Lv,~

= {Yl,

((Y2d34?h)-

- (.h%l,

{Y2d3dhP))dd (hY3d2)d)

+

{hh?~2~Y3d2)

(setting y, = 1 in (12.6)). This completes (iii). Parts (ii) and (iii) yield (iv). Finally, for (v) note that (1.6) shows (y,,y2,y3,%(X)(3)}~ X(Xuy)c$(Xu Y) for any hermitian ideal 2, so by (12.1), s?(X)‘~‘C T&O I 12.8. Open Questions. Are T4 and T, hermitian ideals in the quadratic case? Is T, = E, ? Is T4 = T,? One expects the answer to be No in each case. I 12.9. Remarks. (i) If p E E4 eats tetrads, then so will all y op iff V, eats tetrads (p 0 (y, y,y, y4} E%), and so will all U, y iff U, eats tetrads (U,{Y~Y~Y~Y~~A

(ii) (“,{Y,Y2Y3.hY5kf).

iffp2 eats pentads.

If p E E5 eats pentads, then so will all U, y iff UP eats pentads 1

12.10. Remark. cm 321

In the linear case one has a less useful description

y)>

= b 1 {y,,~2,~3,y,“(y,op)}E~(Xu

because of the ingenious expression (12.11)

JZ( J) = 9L: = L:&3

(.4! the multiplication unital subalgebra [8L:c9L:=

L$9

W,L~L

- LLLJ

algebra generated by all left multiplications, generated by all inner derivations via 2L,L,L,=2L,.,L,+

L02Lb-

9 the D,,.“) L02.b~ L:+ L, and

= &Da,, + L$,, - LAc + L,(,,+ = LB + L,l so

that (12.12) If Z is invariant under all inner derivations of a linear Jordan algebra J, then the largest ideal of J contained in Z is P=L;‘Z= {p~Zl J.(Jep)cZ}= {p~Zl A(J)pcZ}. (The last expression is always the core Z“ inside any space Z, and is con-

186

MCCRIMMON

AND ZEL'MANOV

tained in the middle term; conversely, if I is g-invariant the last two terms coincide since L: p c I =S A(J) p = CBLSp c 91~ I.) An important case of a derivation-invariant space is the set q(J) of values assumed on J by a linearization-closed set P(X) of Jordan polynomials, using (2.9). 1 12.13. Remark. The space E4 is a subalgebra of f(X) since it contains ;;andy isqclosed under U,q: {Y,,Y~,Y~, U,q) = {YO-YO, {y3,p, q),p) 19

2,

o

UpY3)

+

{YbY*?

fJ,Y3d?l

E

LOVP~

+

LfvYI

+

(2 2 $4) c 2 when p, q E E,. The spaces E, for n > 5 do not contain 1 and do not seem to be closed under squaring, but they do form Jordan triple subsystems closed under U,q by {y,, y,, y,, .... y,- , , U,q} = {Yl? {y*,y,,...,y”~,,P;q},P} Yl

o

vJ$~;

uy,f{y..

Yn-1~~~~~

Y4?

- (y,,q, Y39

41

.,)q} - {,$v{B”.~)p}

= U,,J-@VVBP)

-

{Yll

fp}

+ vjf

{Y3,Y,,...,Y,-,,P},Y,,P3 UpY2,

YE-13

...)

+ V/{&&y&l}

+

Y4, Y3, 4) E

- {R”TJ&}

- +9fforp9qEE,,nW

I

12.14. Remark. An n-tad-eater eats n-tads no matter where it occurs. (12.1’)

PEE,=> (~1~2 ~..Y,PY,+rY,4)~9

since {Y,Y~...PY,-~> = {Y~Y~...Y,~~~P)~Y,-, - IY,-,Y,Y~...Y~-~P) Yn-2Y,-11 and iv, -.Y, PY~+ 1~..Y"-2Y,-d = UY, . ..P...Yn--.l {Yn-,"'P"'Y,

Yn-2bYn-l

+

bn~2h

“‘YkPYk+l

“‘Y”-3Yn-,j.

1

13. HEARTY EATERS In the quadratic case eating does not seem to suffice, so we require that our elements eat anything that looks like an n-tad. These hearty tetradand pentad-eating ideals are always hermitian, and the latter always consist precisely of all hearty pentad-eating elements. For all our structural considerations it suffices to deal with elements which eat imbedded n-tads without disturbing the adjacent factors

P(X, 9.**,x,)~j(X)

(13.1)

{Z1...Z,yl..~yn-*p(X)wl.-.Ws}=C(ZI...ZrP1P2P3WI...WsJ

in a(Xu Yu Zu W) for arbitrary r, s and some pi=pi(y,, x,, .... x,) E %(Xu Y). We denote by

.... y,- ,,

STRONGLYPRIMEQUADRATICJORDANALGEBRAS

the linear space of imbedded-n-tad-eaters,

187

and by

I, = (ZEJO its core, the maximal ideal consisting entirely of imbedded-n-tad-eaters. Once more these are linearization-invariant ideals invariant under all endomorphisms and derivations of 9. Note that (13.2)

IE, c E,

since the case r=s=O shows {yl -.Y,-,P) =C {P~P~P~) =P’(X Y) as in (12.1). Again IE,=IEz=IE~=B>IE,~IEs=, ... and Z1=12=13= f > I4 2 Z5 3 . . . are decreasing chains of increasingly voracious eaters. In contrast to eater ideals, hearty eater ideals are clearly hermitian. 13.3. PROPOSITION. All ideals Z, for n 24 are linearization-invariant T-ideals, indeed any formal ideal 2 a f contained in I4 is hermitian. If some yi falls in 2 d j then in (13.1) by homogeneity precisely one pj of each triple of Jordan polynomials has the factor yi, and hence falls back in %; using {...pjp,...} = {...lp,op,...} {...p,pj...> = C (...p;pj...} (p;=l or pk, pj=pjopk or p,) we can move this pj to the right to obtain (for r =O, n = 4, s = m - 4 in (13.1)) for any 2 c E4 the analogue of (12.3): Proof:

V%J-f%

-.w,-4}

c (fy2PW1

-w,-‘$1.

By induction we can remove the Ps one at a time until we get down to an ordinary Jordan 3-tad:

In particular,

x

is m-tad closed { &GE.G~Z}

so Z is hermitian

as in (1.2).

c 2

for all m,

1

Note that we cannot use (13.1) with all the yi E Z tion: even if all yi and p lie in % d j, (13.1) does the pi fall in YP (at least one pi does, but the polynomials in X alone, such as pk = 1). The elements we construct in Section 14 not only

to perform the reducnot guarantee that all other two could be eat imbedded n-tads;

188

MC CRIMMON

AND

ZEL’MANOV

they eat anything resembling n-tads. In particular they will eat non-symmetric associative products y, y, y, y4 p(X). Although this plays no role in our structure theory, it sheds some light on Martindale’s theorem. Since it costs us no extra effort (only extra definitions), we will develop the general theory. A (unital) adic family on a unital special Jordan algebra J is a family of n-linear maps F,, : J” + V into some Q-module I’ for all n 2 1 having the unital Jordan-alternating properties (AI) (13.4)

FR(..., l,...)=F,-,(...,...)

(AII)

F,(...,x,x,...)=F,_,(...,x*,...)

(AIII)

F,(...,x,y,x,...)=F~‘,,(...,

(and therefore by linearization (AII’)

also

F,(...,x,,x,,...)+F,(...,x,,x,,...) = F,p,(...

(13.5) (AIII’)

U,y,...)

{x,x,},

. ..)

F,(~~~,x,,x,,x,,~~~)+F,(~~~,x,,x,,x,,~~.) =

Fn-2(...,

{x1x2x3},

. ..)).

and are compatible with tetrads and pentads

= Fn-3(...,

(13.6) WV)

f’,,(. . ., x19

x27

{x1x2x3x4},...) x3,

X49

x5,

. . .) + F,(. . ., x5,

x4,

x39

x2,

XI,

*‘.

1

only for those Xi E J for which the tetrad {x,x2x3x4} or pentad {x1x2x3x4x5} fulls back in J. (Thus (AIV) is really the special case x5 = 1 of (AV), using (AI).) Note also that (AII) is a consequence of (AI) and (AIII); for non-unital algebras we must assume (AII) and (AIV). Examples of such adic families are the ordinary n-tads T,(x,, ...) x,)=

{Xl.. .X”}

E

H(A, *)

(Jc WA, *)I,

the imbedded n-tads T~~“)(x~,...,x,)={z~...z,x~...x,w~...w,}

E

H(A,*)

for fixed zi, wi in J, or the associative n-ads A,(xl,...,X,)=X~-~.X,

E

A

(Jc A+).

STRONGLY

PRIME

QUADRATIC

JORDAN

189

ALGEBRAS

Note that from any adic family and elements zi, wj in J we can obtain another imbedded adic family by fixing first and last variables, (13.7)

Fpqx,,

...) x,) = Fn+.+.(zl,

...) z,, x1, .... x,, WI, .... ws)

(just as with imbedded n-tads). It is crucial that only outer variables be held fixed! If F is a collection of adic families on f(X), we say p(X) E B(X) eats Y-n-ads if (13.8)

FAY, 3 ..-, Y,-l,P(X))=CF,(P,,P*,P,)EF~‘,(~,8,~)

in V for some pi E 2(X u Y) (depending on p and n but not on F). The case 9 = {T,} of the single adic family of ordinary n-tads gives the ordinary eaters of Section 12; the case where F is all imbedded Tf; %‘Igives us the case of interest of imbedded n-tad eaters. When 9 consists of all possible adic families on y(X) we will call such a p(X) a hearty n-tad eater: it eats anything F,, resembling an n-tad. The theory we develop works for any collection 5 of adic families closed under imbedding (FE F *all F(‘;“‘) E 9). but we will state the theory only for the hearty case where 9 is all adic families on f(X). Once more we denote by

the linear space of hearty n-tad eaters, and by H, = (HE,,)’

its core, the maximal ideal consisting entirely of hearty n-tad-eaters. Again these are linearization-invariant ideals invariant under endomorphisms and derivations of 9. (Linearization-invariance follows usual, and implies derivation-invariance by (2.9). Substitution-invariance under all endomorphisms T of j(X) (fixing Y) follows as usual as long the adic values fall in Vc a (as in our three basic examples) so that can apply an extension of T to (13.8). In the case of general V d 6E must use instead of T(F,Jx,, .... x,)) the induced adic family FT: F;(x,

, .. .. x,,) = F,,(T(xI),

all as as we we

.... T(x,))

(note this satisfies (AIk(II1) since T preserves Jordan products, satisfies (AIV j(V) since T extends to F on 65 so that if (xi . . .x,> then T({x, . ..x.})= T(‘((x, . ..x.})= { T(x,).-. T(x,)}). Here if p all possible F,, it also eats F,T, which means that T(p) eats F,, (T

and E2 eats fixes

190

MCCRIMMON AND ZEL'MANOV

Y, so F”(Y,, *..,Y,-19 T(P)) = I;,‘(YI> ...>Y,-,,P) = CF,T(PI,PZ,PA = C I;,(p;,p;,p;) for pj = T(pi)), so T(p) is again a hearty eater.) We have the inclusions (13.9)

HE,cIE,cE,,

H, c I,, c T,,

with trivially H, = H, = H, = 2 > H, I> H5 3 .... Note that by (AI) the unit 1 is a hearty tetrad eater but not a hearty pentad eater. By (13.3) we see (13.10) all H, for n > 4 are linearization-invariant T-ideals. Because of closure under imbedding, than the ordinary theory in 12.5.

hermitian

the hearty theory runs more smoothly

13.11. HEARTY EATER THEOREM. All hearty pentad eaters are hermitian: the set of hearty pentad eaters forms a hermitian linearization-invariant T-ideal

(0

H,(X)= W(X) a Y(X),

which is not much smaller than the hearty tetrad-eating

(ii)

fW02

(iii)

+ UH~(X)f&(X) HJX)

= H,(X) = HJX) if

= H,(X)

ideal,

4~ @.

Proof: To see that the linear space HE, forms an ideal (hence equals its core H, as in (i)), note that outerness (0.13) PE HE, * U,PE HE, follows as in (12.7) for any adic F from (13.12)

I;S(Y~,YZ~ y3, Y,, u,P) = F,(YI,

{Y,,Y,,Y,,Y,P},Y)-F,(Y,,

+ Fs(Y,, -

Fs(YI,Y~,Y,,

{{Y~,~z,Y~>Y~,Y,P))

(Y,Y~Y},Y,,Y,P)+F~(YI,

{Y,,Y,,

~,Y,,P})

u&d’)

contained in F,(/, %,j) as in (13.8) (note F2 c F3 by (13.4AI) when FU, 9, 9, f, p) = F3V, 9, 3) (noting P E HE, = & = &, so the above pentad

{Y~,Y~~Y~,Y~P)

andtetrads

{{Y~Y~Y~~Y~~Y~P~~

{Y~,Y~~~,Y~,P)

fall in f, so (13.6AIVk(13.6V) are applicable). The identity (13.12) is easily verified in any adic family by expanding all terms up to F,‘s-via (13.4AIII) on the left side, and (AV), (AIV) + (AIII), (AIII), (AIV) +

STRONGLY

(AIII),

(AIII),

~EHE~+U~~EHE~

PRIME

QUADRATIC

JORDAN

191

ALGEBRAS

respectively, on the right side. We have innerness (0.13) from

F,(Y,,Y2,Y3?Y4,

U,Y)=F,(Y,,Y*,y3,y,,P,Y,P) = F:liy.p’(Y, 9Y2,

(by MI)) Y3,

Y4,

PI

(using our feedom to pass up to F, and fix the last two factors to create a new imbedded pentad (13.7) which is eaten by p E HE,) = FU.

2, A

(since p also eats F,). This shows HE, -=I f as in (i). For (ii) we already know H, c H,, and to see U,,pe H, for q E H4, p E HE, it suffices by (i) to show it lies in HE,. To verify (13.8) for n = 5 we use

(13.12)

with

y=q

to

see

that

Fs(~1,y2,~3,~4,

~,P)EF~(%,Y,A.

(Note that (13.6AIV)-( 13.6V) are applicable in this case since { y$yp} ~2 for peE4, {Y$Yqp} = E?p)(6p, 2,9, q) = W’)(%, f, f)

=

($Yfp>cf

+

for

PEEL

and

qEffE4;

here

F5V1,Y,%,qrp)

F,(cfa;, 2, 2, u, 2, P) = Fil’p’(ba, 2, f, 4) + Wp’Lf, f, 8’7 u, 2) = (qE H, a ,J@eats imbedded tetrads ~~pV,Y, Y,cY, H4) c F$‘:P)(y, y,f) U3.7H=1;,Lf5 A AP) = F3V, A 9) (for PEHE~).) The case p = 1 E HE, shows q2 E H, for q E H,. This completes (ii).

For (iii) we have, in analogy with 12S(ii), 2H, c H5

(iii’)

since pEH4*all U,~EH,CHE,, so putting y, = 1 in (13.12) shows via U3.4W that &(Y,~Y,~Y~~Y~~P) = F3(Y1, (Y~Y~YP),Y) F,(Y,> {{y3y2y~,y~P}) + f’4(~,7 {y,y2y}~y~P) + F,(Y,, {y3y2y2P)) - F4(y,, y,, y,, U,p)e F3(y, f, f). (Note (AIV) is applicable since {y2y3yP)>

J’4LF1f,~,

{y3y2y2P)Ef

for

~,P)cF,(~~,~,~,H,)~F,(%,~,%)

PEEL;

here

F~V~~~~~P)+

for

PEH~QA.

Once more, linearizing y --* y, 1 yields 2F,( y, , y,, y,, y, p) E F3(4, f, 9) for all adic F, so by (13.8) and (i), 2p~ HE, = H,. [ The power of pentad-eaters is that when they are annihilated from a pentad they take one of the adversary variables y along with them, whereas a tetrad-eater is annihilated from a tetrad without lessening the number of y’s. Anti-hermitian algebras of Section 2 are abstemious in the sense that they contain no hearty n-tad-eaters for n > 5.

192

MCCRIMMON AND ZEL'MANOV

13.13. Remark. As in 12.13, one can show that HE, is a subalgebra and the HE, are Jordan triple subsystems.

14. CONSTRUCTION OF HEARTY EATERS To show that hermitian polynomials exist, it suffices to prove the existence of a single nonzero hearty pentad-eater in HE5, since then H, is a nonzero hermitian ideal. We will first show how to construct elements of HE4 (the smallest being of degree 16) out of commutators [x, y], and then show how to construct elements of HE, (the smallest being of degree 48) out of commutators [c, , c2] of elements ci of HE4. It is important that this smallest example be a Clifford polynomial as well as hermitian. We get elements of HE, by piling enough commutators [x, y] together; these “ghost” elements act on J, especially in the form D,,(z) = [[x, y], z] as in (0.27). 14.1. HEARTY TETRAD-EATER CONSTRUCTION. Fix x,y~X by 9 the subalgebra of j(X) generated by the range D,,Y(j) derivation D,,,. (i) c3 E D,,(9)’

Zf c1 E D,,jg)‘, CLEF, then [Cc,, c,], c2] E HE,, then [Cc,, c,], c3] E HE,. In particular P&, ~23(4

Y,

and zf also

&,,(w)l, &,,(w)l

Y, z, WI = CCD:,,(Z)~, ~2, ~3) = CCD:,,M2,

~1,

and denote of the inner

D,,,(z2)1~

D:,h)21

are hearty tetrad-eaters in HE,.

(ii)

Zf c E 9 p48= P;‘~E HE, for PIZ(X,

Proof (cf. [32]).

has c 0J+ c2 0JC 9

Y,

z)

=

D,,,.(D:.,(z)~)

then c4E HE,.

=

In particular,

D:,(4~D:,W

Writing

D = D,, for short, we can reduce 9 to its basic generators D(J) if we can find a place to dump the pile Mg of C&multiplications:

STRONGLY

PRIME

QUADRATIC

JORDAN

ALGEBRAS

193

for all adic families as in (13.4) and (13.5) the set of de 9 satisfying Fn(.**d, u*..)EF’,,(...D(.I), J&(a)...)+ F,pI(...&(a).Be) is a linear subspace containing 1 and D(J) trivially by (AI), and closed under U,.d since F,(...U,d,a...) = -F,(...U,a,d...) + F,(...c, (dca}...) (by (NH), (AIII’)) = -F,p,(...V,U,a...) + F,(...d, U,a...) + F,(...c, V’,,,a...) (by (AII’))EF,_ l(...Jtl,(a)...)+F”(...d,~,(a)...)+F~(...c, A&(a)...), so it is a subalgebra and thus contains 9. We will make frequent but tacit use of the fact that for any adic family the F,, are alternating functions of their variables modulo lower F,,,‘s by (AII). Modulo F,(J, J, J) we have (2)

f’&, Y, D,,,(J), J) = 0

since by (AII), (AII), and (AII’), F4(x, y, D,,Y(z), w) = F3(Ux y, y OZ, W) FAU.yy2,z, w) - F,(x, U,(xoz), w) + F2(UxU.vz, w) E F3(J, J, w) (this is clearer if ;E@: 2F,(x,y, Dz, w) = F,(x~y, Dz, w) + F,([x, y]*, Z, W) Fz( I!I~.~,~,z, w)). From this we get (3)

F4(x, 9,9,

J) = 0

since applying (1) twice reduces us to F4(x, D(J), D(J), J), and here F4(x, D,,,(a), D.y,J(bL c) = -F4(x, D,,,(~)~ D,,c(b)y Y) (linearizing (2) and alternation) y + 4: c in F4(x, J, D,,Y(J), y) = 0 from c F4(x, D,,,(J), J, y) = 0 (by (2)). Another linearization x + x, z in this gives for d; 9 that Fdz, D,.,(a), D,,+(b), 4 = -F&, D,,,(a), D,,,(b), 4 - Fzd.‘c, D.,,(4, D,,(b), 4 E FAX, J, 9, 9) + Fdx, g,J, 93)=0 (by (3)), so F4( J, D,,.,,(J), D.,).(J), 9) = 0, and using (1) to dump multiplications onto J we get (4)

F,(J, 9,9,9)

= 0.

Finally we have (5)

F,(J, J, D @a)‘, 9) = 0

using (6)

coJc9

if

CED(Q)

since D(9) 0J= D(Q 0J) - CS0D(J) (by linearized definition (0.12) of derivation) c 9, so for CE D(9), dG9? by (AII), (AII’), (AIII), so F4(J, J,c*,d) = F,(J,coJ,c,d)-F,(J, U,J,d) = F.,(J,coJ,c,d) c F,(J, 9,9,9) = 0 by (4).

194

MC CRIMMON

AND

ZEL’MANOV

To prove (i) it will s&ice to prove the general criterion (i’)

If cl, c2, c3 EJ satisfy F,(J, J, ci, ci) + F,(J, J, c3, c, 0~2) c F3(J, J, J) for all adic F, then [Cc,, c,], c3] E HE,

since then the case c1 ED(C~)~, c2 = c3 l 9 follows [F,(J, J, cl, c,) - 0 by (5), F,(J, J, ~2, cg) E 0 trivially when c2 = c3 by (AII), F,(J, J, c3, cl oc2) =

F,(J, J, ~2, ~1,~2) + F,(J, J, ~2, ~2, ~1) = FdJ, J, Uc2c,) + F,(J, J, c:, ~1) - F4(J, J, cz, c,)=O by (AII’), (AIII), (AII), (5)], as does the case cl, c,~D(g)~, c,E~@ [F,(J, J, ci, cj) -0 since either i=j=2 and we get -0 by (AII), or at least one,of ci, cj falls in D(B)’ and we get = 0 by (5), and F,(J, J, ~3, cloc2)r0 by (5)]. To prove (i’) let us first note in general that since (JJc, ~2) c J from the special case F4 = T4 of the hypothesis in (i’), we can apply (13.6AIV) to see via (AII’), (III’) that (7)

f’,(a,

b, EC,, c,l)=

F,({a,

b, ~1, c2} -c2+,,

6, a),

+F2({~1,b,~},c,)+F,((c,,~,b},c,)

- FAaob, ~2, c,). Thus modulo F,(J, J, J) we have F~Y,>Y,>Y,,

CCC,,~219~31)

= F,56(y1,y2,y3, [cl, c21~c3)-F6(y1~y2~y3~c3~ ccl, ‘21) = F$“‘c3)(y2,y3, cc,, c2l)-Fc,(y,,

{y2y3c3),

cc,, c21)

+ F$-“‘,c3;1)(y3,y,, Cc,, ~23) = [@q{c,

y,y,},

c2,+F’2y’;c3)({c2Y2Y3}, Cl)

- F3”‘;c3)(y20y3, c2, cl)] -0+ + f'2"-'3:')({c2 y, y,}, cl)-

[F$~“‘*L.3;1)({c1 y,y,},

~2)

F5-"1*"3;l)(y~oy~,c2, cl)]

(by (7) applied to imbedded tetrads (13.7), since F,(J, J, Ci, Cj) - 0 by hypothesis and F~y:‘)=OmodF,(J,J,J))= [O+O-F5(y~,y20y3~c2~c~7c3)] + [O+O-F5(yl, c3,y20y3, c2, cl)] (since F$“3’(J, ci)- F$~“~““‘(J, Ci)-0 from our hypothesis that F,(J, J, cjr cj) s 0) = --F,(Y,,~,oY,, -Fdyir

{c~c~c~})+F~(~~,y~~y~,

c3~(~2~y3)9 ~2, c,)+F,(y,,y,~y,,

(by (AIII’),

(AII’))

~3, cl, ~2) ~3, ~2, ~1)

195

STRONGLYPRIMEQUADRATICJORDANALGEBRAS

= - ~s(Y,,Y,~Y,,

c3,

Cl,

c*)+1;5iYl~Y2°1’3~

c3,

c2,

Cl)

(by Fd(J, J, Ci, Cj) = 0) = ~4(Yl,Y,~Y37 E 0

c3,

ClOC,)

(by (AW)

(by hypothesis).

Therefore by (13.8), [Cc,, c2], c3] E HE,. For (ii), we have F4(c4, J, J, J) E F4(c2, c20 J, J, J) = F4(c, c20J, COJ, J) c FJGS, 9,2?, J) (by hypothesis on c) - 0 (by (4)), so c4 E HE, by (13.8). By (6) any c E D(g) has COJc 22; if c E D(D(g)‘) c D(g) then also ~~01~62 since c is a sum of elements D(I)(d)2)=D(d)oD2(d)=c, oc2 with CUED, and c2 a sum of elements (cl 0~~)~ and (c,oc2)od (dE9), and whenever c,~J+c,~J+(c,~c,)~Jc~ we have {(c10c2)od}~J = - v,,vdvc2 + VcIcczVd + v,., dVc, + V,,Od~Vrz}J and = V,., c2,c,,,c2 J = { v,., v,.,,., //c2+ v,., 1/q,,, .,r’z - u,.,,,, vc, 9.,> Jv where V,,J+ VC,yC2J~ ~2 and VczLd, VC,0d, UC,,L.2map GS into 9. In particular, we can take d = D(z), c = D(D’(z)~) = D2(z) 0D3(z). 1 wLvJq (cpc+J

Note that the argument of 14.1 needed only axioms (AI)-(AIII) for adic families. Once we have hearty tetrad-eaters, we can commute them to get hearty pentad-eaters; for this we will need axioms (AIV), (AV) for adic families. 14.2. HEARTY PENTAD-EATER

CONSTRUCTION.

Zf~(x~,...,x,,x)~%(X)

is a Jordan polynomial of the form

(0)

P(X, 9...>X,, X)=Cc([Xi”‘Xi,[Xi,

X] Xj, “‘Xjx

in t%(X), then 0)

de 1, .... e,,f)EHs

foralle;EHE,,

fEE4.

In particular,

(ii)

C[HE,> E41, HE41 = H,

(iii)

UCM~,E~I HE4 c Hs

so tf cl, c2, c3E HE, are hearty tetrad-eaters we get hermitian polynomials [Cc,, c2], c3] and [c,, c2]’ in H,. In particular, we have a hermitian polynomial

which is also a cltfford polynomial: qys# 0 on H3(@), hence q$ # 0 on nonzero doubly interconnected i-special algebrasfor any n.

196

MC CRIMMON

AND

ZEL’MANOV

ProoJ: By (13.1 l), H, = HE, consists of all hearty pentad-eaters, need only show that our elements fall in HE,, i.e., by (13.8) that

so we

F,(J, J, J, J, PI = FdJ, J, J)

(*I

for all adic families F. To make use of the form (0) of p involving commutators, we need to be able to “replace the dots by commas”; this follows from the generalization of ( 13.4AI k( 13.4111) (A)

If

q(X) E y(X)

is a Jordan polynomial of the form . . . xg in a(X), then for any adic family F and any yi, zj we have F,+,+ ,(Y,, .... yr, d-V, ~1, .... z,) = C a,Fr+s+n(~,r .... yr, xi,, .... xi,,, 21, .... z,).

4l-U

=

CI

@-lxil

The set of all q satisfying (A) is a Jordan subalgebra of y(X) (which trivially contains all XE X, and hence must be all of y(X)): it is a linear space containing 1 and closed under qpq, since when p = CJ BJ yj, . . . yjm we have for any F that F,(qpq) = F,(q,p, q) (by (AIII))=F’,‘;P,4)(q) = C ~,F(‘;P.q)(x. I,, .... xi,) (applying (A) to q and the imbedded adic family pm;, = C cr,F, + 2(x;,, .... xi”, p, q) = 1 CI,F(~XII-~Q~)(~) = 1 a, C fiJ F~r+;q)( yj, , .... yjm) (applying (A) to p and P~yz~~~~.,x+))=

C a,PJFn+m+

i(Xil, ...pXi”, Yjl, ...yYjm, q)

=

C a,PJF(ir13...*‘v”(q)

=

C aIflJaI’ F~‘I----“~~;~‘(x~~, .... xc) (applying (A) again to q and the imbedded F)=Ca,PJtll’Fn+m+n,(~i,, .... xi,, yjI, .... yjm, xii, .... xi;). This is just condition (A) for qpq when r = s = 0. The case of general r, s then follows from

this by applying it to the imbedded adic family ~Y1....,Yr;rl,....rs).(Despite appearances, property (A) follows directly from axioms (AI)-(AIII) and involves the concept of imbedded adic family only as a technical aid.) In view of the form (0) for p and the result (A), to obtain (*) we must show (**)

F6+r+s(J, J, J, J, ei,, .... ei,, [ei,f],

ejl, .... ej)c F3(J, J, J)

(where we agree F( [x, y]) means F(x, y) - F( y, x), since in general [x, y] does not lie in J). For this it suffices to show (1)

Fs(J,J, J, J, [HE,, &I I= F&t J, 4

(2)

F3 + .(J, J, J, HE,, .... HE,) = F,(J, J, J)

since then each term F6+r+s(J,J,J, J, cl, .... c,, [e,f],d,, .... d,) in (**), (ci, di, e E HE,, f E E4) will reduce to F’~‘~r~YI~dl*...*~(J,J, J, cl, .... c,) c FSJ:[e~fl,dl.-.,d$J, J, J) (by (2) for the imbedded adic family) = F~;dl*-.-ds)(J, J, J, J, [e,f]) c F$$;dl,-*ds)(J, J, J) (by (1) for the imbedded family) = F3 + .( J, J, J, d, , .... 4 = FAJ, J, J) (by (2) for F).

STRONGLY

PRIME

QUADRATIC

JORDAN

197

ALGEBRAS

Thus (*) reduces to (1) and (2). Establishment of (2) follows easily from the definition of hearty tetrad-eater: the C,E HE, can be removed from F3 +JJ, J, J, Cl, .... c,) one at a time from the left without disturbing the remaining c’s, as ci eats the preceding imbedded tetrad F n+4-i(J, J, J, ci, .... c,) = F&‘;C~+l---Cn)(J, J, J, ci) c F~‘~~~+~~-~‘~)(J,J, J) (by definition of ci E HE,) = F, + 4 ~ (i+ , ,(J, J, J, ci+ i, .... c,) until we reach F n +4P Cn+ ,,(J, J, J) = F,(J, J, J) when i = n. So far we still have used only the axioms (AI)(AIII). It is only in establishing (1) that we need (AW, (AVJ. To prove (1) we calculate

F6(~19~29~3T~4T Ml) :=F6(y,,y2,Y3,Y4,e,f)--6(y,,y2,y3,y4,f,e)

=F~(Y,, +F3(yl,

EF,(J, J, J)

(e,y4,y3,y2) {.h.hy4~

{ JJJ~}

{e,Y4,Y,,Yd)

{Y2ry3,Y4,e,f))--3(y,,f,

4-F4h

{f,Y2yY3yY4},

e)

{Y4Y3Y2)&

(note (13.6AIV), 13.6V) are applicable here and (~,Y~,Y~,Y~) lie in J for e,feE4, e,f}

=

since

and

ek T$“f’(J, J,J) (for eM&)=

T!p(Y2~Y3~~4~

c J (for f~ E4); h ere F,(J, J, J, e) c F,(J, J, J) for e E HE,).

This

establishes (1). Thus (i) holds; in particular (ii) and (iii) hold so q4* is hermitian. To see that it is also Clifford, note that if JI H3(@) with symmetric matrix units uii = E, + Eji then one caluclates

so that q4d”13?

u23~ =

u23~

u,3;

[[P16t”13,

u23~

hi(“12v

= CC%,,

u12~

Ul31,

u23~

u23?

u23~ u23,

U23r

u13),Pltd”,2~

u12;

u12~

U23r

U23r

u23~

u23~

%2)

u,2)1~

42)1

u131=u12

so that 4:; = (u:~)~ = (e, + e2)m = e, + e, and hence q& # 0 if JI H,(Q). Once q& is Clifford, by (7.15) no nonzero doubly interconnected i-special algebra can satisfy it. 1 The polynomial q4* may be intractably improvement on the original polynomial

large, but it is at least an

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MCCRIMMON

AND ZEL'MANOV

Already ps(x, y, z) = D,,,(D,,,(z)*) is nonzero on degree 3 3 algebras. It would be interesting to know if ps E HE5 or HE4. Thus we have our first example of a Zel’manov polynomial, one which is simultaneously hermitian and Clifford. For applications to Martindale’s theorem [22] it is useful to work with a hermitian ideal slightly smaller than H,. The ideal (14.3)

Qdm

a Y(X)

is the linearization-invariant T-ideal generated by the polynomial q.,*; since Q&X) c H,(X) c H4(X) c&(X) we have by (13.3) that the ideal Q&X) is also hermitian. Because qd8 is built out of p16, which is built out of commutators D,,Y(z) = [[x, y], z] = [V,, V,] J, we see (14.4)

Q48(X)~~~(C~,8,~1)~~~(~~~).

For our structure theory in Parts I and II we did not need to know about the internal structure of hermitian or Clifford polynomials, but for Martindale’s theorem it is important that the internal structure of qd8 guarantees that [[J, J], J] and Jo J generate J for any “hearty” algebra. 14.5 Qd8 IDEAL THEOREM. The ideal Qds (X) a y(X) is a linearizationinvariant hermitian T-ideal generated by qd8. If an i-special algebra J is doubly interconnected, then J= Q4*(J). Whenever J= Qd8(J) we have (i) (ii) (iii)

J= Q,,(J)‘“‘for J=&(JoJ)=Y~([J,

all n; J, J]);

J= H,(A, *) is ample in any *-envelope JC H(A, *).

Proof J= J/Qd8(J) inherits i-speciality and the double interconnection idempotents 2, of (7.15), and now by construction satisfies the Clifford identity qd8E Q4&X); by (14.2) this forces e, = 0 in J, hence J= &(.Z1) = 0 by definition of interconnection, so J = Qd8(J). Now assume that J= Q&J). Then by specializing (14.4) we get J=Q&J)cY~([J, J, J])csi,(JoJ)*J=$([J, J, J])=Y,(JoJ)asin(ii). In general (ii) implies (i) by induction (if J= Q(“) then also J=~~([Q”‘) Q’“’ Q’“‘])c Q(“+l’ since [K, K, K] c {KKK} c K(l)). Alternately, (i) follow; since in general

(14.6)

J = X(J)

for hermitian

X =s.J = 2?(J)‘“‘;

indeed, any hermitian polynomial is (by linearization-invariance (13.4)) a sum of homogeneous hermitian polynomials of degree >3 (there are no homogeneous hermitian polynomials of degree < 2, i.e., x, x2, x 0 y are not

STRONGLY PRIMEQUADRATICJORDANALGEBRAS

hermitian), so J=~(J)c~~~J=J(‘)~J=J(“) Finally, (iii) holds by (1.3) whenever J= x(J)

199

for all n, SO J=x(J)‘“‘. for hermitian 2. 1

Note that because qd8is of such high degree it would be difficult to prove directly that q&J) # 0 (much less than Q&5) = J) when the idempotents are merely interconnected (e,= C U,y Y,~ for some unknown number of X~E J,, yjje J,j), whereas in the strongly connected case we can calculate directly as in (14.2). There is another way to see that J= Q&J) * J= H,(A, *), making use for the first time of the omnivorous nature of Q4,: becauseQd8c H4 eats all possible 4-ads it eats the associative 4-ads A,(y,, y,, y3,p(X)]-= y,y,y,p(X)=Cp,p,p,ina(X), thus when J=H,(J) we have JJJJcJJJ, and the associative envelope A generated by J is already generated by the terms of length 6 3. But this condition always guarantees that J is ample.

14.7. PROPOSITION. Zf A is a *-envelope for J such that A = j.?J, then J= H,(A, *) is an ample subspace. Any a E A can by hypothesis be written as a = Ci XiyiZi for xi, yi E j and zi E J; then a + a* = C {xi yizi} c { jgJ} c J, so J contains all traces. It contains all norms since aa* = xi xi y,z: y,x, + Cizj xi yizizj yjxj = Cj U,U,,z: + Cicj (xi yizizj yjxj) + (xi yizizjxj)* c J+ tr(A)c J by the previous part. (Or: J is n-tad closed in A for all n since xi . ..x., = C Pi4iri* Ix1 e-.x,} =C {piqiri} E J.) Th us J is an ample subspace of WA, *I. I ProoJ

Thus ampleness and hermitianness are closely related to generation by length 3, which in turn is closely related to eating associative 4-ads into 3-ads. The eating of assymmetric 4-ads A4 is a very strong property, and immediately yields the eating of ordinary symmetric tetrads T,. The general version of Martindale’s theorem says that Jordan homomorphisms may be extended from JC H(A, *) to A when J = Q&J) is unital; by 14.5 this holds when J is doubly interconnected, which is essentially Martindale’s original result. For this result it is crucial that everything would work equally well Q&F+CcA %, fl)-J%Q): for any hermitian ideal with this property. We conjecture that this holds for H,, or more generally for any hermitian ideal: (?)

% hermitian =- Y?(X) c Y8( [I, PART

IV.

f, 91).

CLASSIFICATION

Once we know that hermitian Clifford polynomials exist, we can amalgamate the Hermitian and Clifford Structure Theorems to obtain a

200

MCCRIMMON

AND ZEL'MANOV

.

description of all strongly prime algebras, and very precise descriptions of all simple and division Jordan algebras. In particular, we obtain an idempotent-free classification of all finite-dimensional simple Jordan algebras which are i-special. We first break strongly prime algebras into albert and i-special algebras, and then analyze the i-special ones.

15. CLASSIFICATION

THEOREMS

Here we obtain the main structure theorem for strongly prime algebras, from which the structure theorems for simple and division algebras quickly follow. The Prime Dichotomy Theorem splits strongly prime algebras into albert forms and i-special algebras, the Hermitian Dichotomy Theorem splits the i-special algebras into hermitian forms and anti-hermitian algebras, and the Zelmanov polynomial shows that anti-hermitian algebras are Clifford forms. The original Dichotomy Theorem [29,9] applied to imbeddable primes, and since all nondegenerate algebras are imbeddable we have (recalling that strongly prime = prime + nondegenerate). 15.1 PRIME DICHOTOMY THEOREM [9, 5.11. algebra is either i-special or an albert form. Combining

A strongly prime Jordan

this with our analysis of i-special algebras yields

15.2. STRONGLY PRIME STRUCTURE THEOREM. A strongly prime Jordan @-algebra is a clifford, an albert, or a hermitian form: either there is a unitalscalar extension s= I% for an algebraically closedfield a such that J is cltfford or albert, (I)

J=fi=H@)

ts ’ hermitian and tiny cltfford of dimension1;

(1% )

f= J@, 1) = H@) is hermitian and small cltfford dimension3; 2 = J( &, 1) is large clifford of dimension 2 4;

(E)

? = J(fi, 1) is albert of dimension 27;

(I?)

of

in which case the central closure J= f(J) -I J is simple, and is cltfford or albert or is sandwidchedamply between two hermitian and cltfford algebras over the field 3 = r(J),

(I)

J= H,(,/%)

is a division algebra $1 4 J= 01 + fl (a a 6subspace of @, so L? = 8 if char # 2);

1

STRONGLY

PRIME

QUADRATIC

JORDAN

(III)

J(&,, 1) + J+ J(&,, 1) for Qi nondegenerate forms over fields 8i, 6 c 6-O c S1 c fl; J= J( Q, 1) is simple and clzfford over 8;

(IV)

J= J(N, 1) is simple and ulbert over 6;

(II)

201

ALGEBRAS

quadratic

or else J consists of hermitian elements

(V)

%(A, *l=*(J)

aJcH(Qb4,4

lying between a *-prime associative @-algebra A and its *-prime quotient ring Q(A), where A has P.I. degree >3 (satisfies no cltfford identity). Proof By the Prime Dichotomy Theorem 15.1 we know that a strongly prime Jordan algebra is either an albert form as in (IV) and (IV), or is i-special. By the Hermitian Dichotomy Theorem 3.3 the i-special prime algebras are of either hermitian or anti-hermitian type By the Hermitian Structure Theorem 3.4 the algebras of hermitian type are hermitian forms as in (V) (recall that nondegenerate implies hereditarily semiprime). By 14.2 the anti-hermitian algebras J satisfy the hermitian Clifford identity qd8 E 0. By the Clifford St ructure Theorem 11.1,:’ is a Clifford form with a Clifford extension J= I;lij= J(Q, 1) as in (i’)(III) (dim J’= 2 is ruled out when 7 is prime and d is algebraically closed), hence by the Clifford Form Theorem 6.1, J has the form (Ik(II1). i

In view of Pchelintsev’s examples [24] of pathological degenerate prime algebras, the restriction to strongly prime algebras is a natural one. 15.3. Remark. In fact [31], all such algebras are prime. The only trouble is with the hermitian forms (Vtin all other cases J has a nondegenerate prime scalar extension T= a$ which forces J itself to be nondegenerate and prime: if 0 #I,, I, a J have UI, I2 = 0 then the scalar extensions 0 # s”i1,, di, a fij would be orthogonal, contrary to primeness, and if ZE J were trivial then it would be trivial in fij, contrary to nondegeneracy. 1

As a consequence of the structure theorem we see that a nondegenerate prime Jordan algebra which is i-special is actually special. By [33; 27, Corollary 4, p. 2555; 38, Theorem 2.11) any nondegenerate algebra is a subdirect product of strongly prime algebras, so 15.4.

Any

PROPOSITION.

actually special.

nondegenerate

algebra

which

is i-special

is

1

It would be desirable to have a direct proof of this, so the Prime Dichotomy Theorem 15.1 would directly assert that a strongly prime algebra is albert or special.

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MCCRIMMON

AND ZEL'MANOV

The Prime Structure Theorem leads quickly to an exact description of all possible simple Jordan algebras of arbitrary dimension. Simple algebras are always prime, and absolute zero divisors are easily seen to generate nil ideals (indeed, all of Z(J) is nil), so simple unital algebras are nondegenerate. With more sophistication one can remove the unitality condition: absolute zero divisors generate locally nilpotent ideals (by [31], [34] for linear Jordan algebras, and by recent work [6] of Kostrikin for general algebras-it would be desirable to have an easier proof of this), and simple algebras are never locally nilpotent. Furthermore, for simple algebras the centroid T(J) is already a field and therefore J is centrally closed (J=T(J)-’ J=J and $= r(7) = T(J)). In this case the Prime Structure Theorem tightens up to yield the following 15.5. SIMPLE

STRUCTURE

THEOREM.

A simple Jordan @-algebra is

isomorphic to one of

(I) an ample subspaceH,(A, *) in a hermitian algebra H(A, *) for a *-simple associative @-algebra A, which thus has one of the two forms (Ia) A + for simple A, (Ib) H,(A, *) for simple A with involution; (IIa) an ample outer ideal in a small cltfford algebra JK?,, 1) -9 J-+ J(Ql,

1)

for nondegeneratequadratic forms Qi over @-fields Qi, LkQocQ,cfi,

12= r(J);

(IIb) a large cltfford algebra J(Q, 1) of a nondegenerate quadratic form L2 over a field L2I @; (III) an albert algebra J(N, 1) of a nondegenerate admissible cubic form N over a field Sz3 @, which thus is obtained from the First or Second Tits Construction WIa)

J1(A,p)

= AOAOA,

N(a,Oa,@a,)

= n(ao) + /da,)

+ t.i’n(a*) for A central simple associative of degree 3 over Sz, 0 # p E52,

(IIIb)

J,(A, *, u, p) = H(A, *) 0 A,

N(ao 0 al)

= n(a,)+

pn(a,) + p*n(a*) for A central simple associative of degree 3 over Y with involution * of second kind, Y separable quadratic over Sz= H( Y, *), UEH(A, *) with n(u)=pp*#Ofor pEO. Proof As we noted, a simple unital J is prime, nondegenerate (by [31,6]), and equals its central closure, and thus has one of the forms (Ib(V) in 15.1. In case (V) of 15.2, H,(A, *) d J forces H,(A, *)= J by

STRONGLYPRIMEQUADRATICJORDANALGEBRAS

203

simplicity, and we have our present case (I). Note that whenever J= H,(A, *) is simple we can replace A by a *-tight cover and assume that A is *-simple; then either A is simple as in (Ib), or A = B H B* for simple B, in which case the only ampie subspace is H,(A, *) = tr(A, *)= {(b, b*)} E B+ as in (Ia). In the division algebra case (I) of 15.2 we have J= J= H,,(A, *) where A = 6 is a field with trivial involution, which falls under case (Ib) here. The small case (II) of 15.2 falls under (IIa) here. Since we noted J = J, @, the large Clifford and albert cases (III), (IV) of 15.2 reduce to (IIb), (III) here. It is known [12, Theorem 10, p. 3131 that the albert algebras are obtained by the First or Second Tits Constructions (O.lOI), (0.1011) as in (IIIa), (IIIb). 1 Conversely, it is known [2] that all albert algebras (III) and all large Clifford algebras (IIb) (1;2nondegenerate of dimension >3 over a field) are simple. In (IIa), J( Q, 1) is simple, and hence [ 181 so is its ample outer ideal J. It is known [17] that H(A, *) is simple for all unital *-simple A (hence by [18] for any ample outer ideal H,(A, *) as in (I)), but it is an open question whether this remains true for all A. Note in particular that our methods provide an idempotent-free approach to the structure theory of i-special finite-dimensional algebras. (However, the present proof [29, 21 that i-exceptional algebras are albert brings to bear the entire idempotent-laden classical theory.) The considerations in the Clifford Form Theorem 6.1 yield a shorter proof of Osborn’s Capacity Two Theorem: the capacity two matrix algebras H(D,, *) where D is not commutative are subsumed (by the Clifford Interconnection Theorem 7.11) under the hermitian theory, and the only capacity two algebras we have to examine are those which we already know live inside Clifford algebras 2. A detailed analysis is still needed in characteristic 2, since Clifford algebras (unlike albert algebras) are very sensitive to scalar extension: they are not necessarily outer-simple or outer-central, and we saw there exist phenomena in J which disappear in dj. Finite-dimensional semiprime algebras always have units, so there is no problem about nondegeneracy or sufficiency of the conditions. 15.6. FINITE-DIMENSIONAL STRUCTURE THEOREM. The simple Jordan algebrasfinite-dimensional over a field @ are precisely all (Ia)

M,,(A)+ for a finite-dimensional division algebra A over @;

(lb) HoWn(A), *I f or x* =X’ involution d + d on A;

(IC)

H,(M,,(B),

of transpose type relative to an

*) for x* =sx’sC’ of symplectic type, s’= -s;

204

MCCRIMMON

(IIa)

AND ZEL'MANOV

amply sandwichedsmall cltfford algebras J(Q,, l)-+JJJ(Q,,

1);

(IIb)

large clifford algebras J(Q, 1) for a finite-dimensional nondegenerate quadratic form Q over a finite extension field IR of @;

(IIIa) J,(A, u) for central simple A of degree 3 over a finite extension field Sz of @, O#p~iIi?; (IIIb) J,(A, *, u, u) for central simple A of degree 3 over Y, Y quadratic over a finite extension field 52 of @, u E H(A, *) with n(u) = up* # 0 for * an involution of the second kind on A. Proof In 15.5(I) the Wedderburn Structure Theorem says A = M,(A) consists of n x n matrices over a division algebra A, where the involutions are of either transpose type (as in (Ib)) or symplectic type (as in (1~)). 1

Since division algebras are always simple and unital, we get a similarly precise description of them. 15.7. DIVISION ALGEBRA

STRUCTURE

THEOREM.

A Jordan algebra is a

division algebra iff it is isomorphic to one of

(Ia) (Ib) (IIa)

A+ for an associative division algebra A; H,(A, *) for an associative division algebra with involution;

an ample outer ideal J + J( Q, 1) in a small Clifford algebra of an anisotropic form Q over afield Sz;

(IIb) a large Clifford algebra J( Q, 1) f or an anisotropic quadratic form Q over a field Sz; (III) an albert division algebra J(N, 1) for an anisotropic admissible cubic form N over a field IR, hence obtained from the First or Second Tits Constructions WW pLnn(A);

Jl(4 pL)for an associativedegree 3 division algebra A over a,

(IIIb) Jz(A, *, u, u) for A an associative degree 3 division algebra over !P with involution * of second kind, ‘Y separable quadratic over 52=H(!P,*),u~H(d,*) withn(u)=uu*foru#n(A). Proof We can apply the Simple Theorem 15.5. In case (I) of 15.5, the Herstein-Osborn theorem asserts that if H,(A, *) is a division algebra for a *-simple A then either A = A is already a division algebra (as in (Ib)), or A = A ERAop with exchange involution (as in (Ia)), or A = M,(Q) is a split quaternion algebra with standard involution and H,,(A) = 01 (which falls

205

STRONGLYPRIMEQUADRATICJORDANALGEBRAS

under case (Ia) for A = Sz). Conversely, if A is a division algebra so is A + and H(A, *) (hence by [lS] any ample H,(A, *) is too). In case (IIb) of 15.5, the ample outer ideal J+ J(Q, 1) is a division algebra iff J(Q, 1) is [ 181, and in (IIa) of 15.5, J(Q, 1) is a division algebra iff Q is anisotropic. These give us the present cases (IIa), (IIb). In case (III) of 15.5, J(N, 1) is a division algebra iff N is anisotropic, and anisotropy of N in (O.lOI), (0.1011) means A = A is a division algebra and p 4 n(A) [ 11, Theorem 6, p. 507, Theorem 7, p. 5091. This gives the present case (III). 1 This extends the division algebra classification of [30] to characteristic 2, using the more powerful techniques [32] of Section 1 developed for prime algebras instead of the result

(which still has not been verified over general 0 when 4 $ @!).

PART

V.

RADICALS

Here we draw some of the surprising consequences that the structure theory for prime algebras has for the free algebra. As in alternative algebras (but unlike associative algebras), the universal free Jordan algebra has a radical. This “universal” or “generic” radical consists of all radical identities. It coincides with the nondegenerate, nil, strictly nil, and Jacobson radicals of the free algebra, thus these radicals coincide “generically,” and it is only in particular algebras that they can differ. Moreover, we show that the nondegenerate and strictly nil radicals never differ for P.I. algebras, using quadratic absorbers of inner ideals to pass from special P.I. algebras to general ones.

16. RADICAL IDENTITIES Radical identities are free Jordan polynomials that take on only radical values. They are just the elements of the radical (nondegenerate = nil = Jacobson) of the universal free algebra, and are precisely the identities which vanish on the standard Jordan algebras (the albert algebras and the special algebras). Our structure theory will imply that the nondegenerate,

strictly nil, and

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MCCRIMMON AND ZEL'MANOV

Jacobson radicals coincide “universally”, s-identities. We let

and consists precisely of all albert

alb( X) denote the T-ideal of all u-identities (albert identities) in a universal set X of indeterminates satisfied by all albert algebras (equivalently, by all split albert algebras over fields), and by

the ideal of all s-identities (special identities) satisfied by all special algebras (equivalently, by the free special Jordan algebra FSJ(X)). 16.1. RADICAL

IDENTITY THEOREM [32]. precisely the albert s-identities: if q = FJ(X) algebra (X infinite) then

Rad(%!) = Nil(@) = SNil(4)

= U(a)

The radical identities are is a universal free Jordan

=&lb(X)

n P’(X).

Proof. [32; 10, Theorem 8.111. Writing a=a lb(X), Y==(X) we always have ~?(a’) c SNil(%!) c Nil(%) c Rad(%) for any algebra %!, and Rad(‘B)cLZ nY since the values taken on by Rad(%) on any algebra form a q.i. ideal, and the free special and split albert algebras have no q.i. ideals (the only quasi-invertible elements of FSJ(X) are constants, and a split albert algebra H(M,(O)) over a field is simple and unital), so Rad(%) vanishes on such algebras and Rad(%) c a n Y. Finally, we have GEn Y c Y(a) since the nondegenerate algebra J= %/Y(B) is a subdirect product Jz j-I Ji of prime nondegenerate algebras Ji [27, 191, which by the Prime Dichotomy Theorem 15.1 are albert or special so that either a(J,) = 0 or 9’(Ji) = 0, therefore (a n Y)(J,) = 0 in either case, and consequently an Y vanishes on JC n Ji, and (an S”)(e) c Y(42). a

16.2. Remark. Various parts of the above chain of equalities can be established without reference to the structure theory, depending only on universal freedom. For Rad(%) = Nil(%) we need only show that Rad(%!) is a nil ideal, and here for any p E Rad(‘%) we can choose x in X not appearing in p by infinitude of X, since Q is graded with respect to x the element U,p E Rad(!#) is nilpotent by Amitsur’s Graded Trick [ 151, [2,4.5.13], hence p is nilpotent in %! under a homomorphism fixing p but sending x to 1.

STRONGLY

PRIME

QUADRATIC

JORDAN

ALGEBRAS

207

The equality Nil(%) = SNil(%) holds because Nil(%) is strictly nil: any element q =pl o1 + . . . +p,o, E 4YQ for pie Nil(%), ORE Q remains nilpotent since if we choose xi, xii not appearing in infinitude of X again) then PI) s3.9P, (by x.. is again in the ideal Nil(%) and is therefore PC Ci up,xf + Cicj up,,p, r/ nilpotent, so under the homomorphism @ + u?& sending pi -pi, xi --+ oil, xii + oiojl it goes into a nilpotent element C U,,o: + C U,,,,,wiwj = C,p?o? +Cicjpiopj-oioj= (Cp,o,)’ =q2, so q is nilpotent. (If $E @ then q itself is the image of xpioxi under xi + wi) The final equality Nil(@) = Y(e) is the trickiest. The idea is to show that fl= Nil(&)/Z(%) is strictly nil of bounded index d= 2: p2 E 0. Since quadratic identities always extend, it suffices if p* = 0 for each jj E R, i.e., if p* E Y(e) for all p E Nil(%). Because %!/Y(%!) is nondegenerate, it s&ices if Up24!Lc Y(e), i.e., if p E Nil(%) * U, U,C& c 5?(%).

(*)

l Nil(%) and choose x in X different from the x;s (by infinitude again). Then U,XE Nil(@) is nilpotent of some index d= d(p), which is then universal for the inner ideal Up4’& in any scalar extension sQ : Fix p(x,, .... x,)

(upU)d=o

for all u E %a

since ( Up~)d= 0 and there is a homomorphism sending x to u. The inner ideal

generated b3d= (U,b)dE

@ + +& fixing p and

by

p has u, %2 = up % (by (0.2), (0.3)), so (U,4&)“= 0 for all be B,, and B is strictly nil of bounded

index. Then by the Bounded Index Theorem (cf. [35], (16.3)

B = Y(B)

[ 19, Theorem 3.11)

if B is strictly nil of bounded index,

so B is degenerate. Now we have a general Inner Transfer Principle Lemma 15; 10, Lemma 8.121: (16.4)

If B d J is inner then 17,9(B)

In our case we have U,U,43!cUsB=U,9’(B) 16.4)), so (*) holds. i

[31,

c 5?(J).

(by (16.3)) cU(%!)

(by

208

MC CRIMMON

16.5. Remark. (16.6)

AND

ZEL’MANOV

The above radical result anY=2ye)

was used originally in an ingenious bootstrap operation [32] to boost the Dichotomy Theorem 15.1 from strictly-nil-free algebras to nondegenerate algebras, before the nondegenerate imbedding theorem [S] was noticed. To see how this works, note a n Y 2 SNil(%) = L?(e) follows as above, and the Old Dichotomy Theorem [29; 9; 2, Proposition 7.31 applied to the strictly-nil-free algebra %/SNil(&) gives 05 n Y c SNil(Q), so once more we have a n Y = SNil(%!) = Y(a) as in 16.1. Suppose now that J is prime but only nondegenerate, not necessarily strictly-nil-free. Then a n Y vanishes on J since by 16.1 its values fall in (a n Y)(J) c Y(J) = 0 by nondegeneracy of J. Thus a(J) and Y(J) are ideals m J with UncJ, Y’(J) = (U&Y)(J) c (a n Y)(J) = 0 (note that U&,....a.) db, 9 ...Y6,) is a value of ~~~x,,...,xn~g(x,+ ,, .-, x,+,)E UaY for f Ea, g EY by infinitude of X again). Thus by primenessof J we must have either a(J) = 0 or Y(J) = 0. We are trying to establish the dichotomy that J is either i-special or albert; clearly Y(J)= 0 iff J is i-special, and we will see that a(J)=0 implies SNiI(J) = Y(J) = 0, so that by the Old Dichotomy Theorem J is i-special or albert in this case too. Indeed, it suffices to establish the general result (16.7)

~(J)=O*SNiI(J)=~(J).

We will see later in (17.3) that any polynomial identity forces SNil(J) = Y(J), but in the present situation we can give a somewhat shorter proof. We will prove s= SNil(J)/Y(J) is zero by proving that it is strictly nil of bounded index (hence degenerate by ( 16.3)), yet an ideal in a nondegenerate algebra J/Y(J) (hence itself nondegenerate). Thus everything reduces to showing S is strictly nil of bounded index. Clearly it is nil with a(S) = 0 and Y(s) = 0, and the same holds for s[ T], so for (16.7) it suffices to prove (16.8)

a(J) = L?(J) = 0 * all nilpotent elements of J have index < 54.

Indeed, an albert algebra A satisfies (16.9)

Sd T, , a.., Ts4) = 0 for all multiplications

Ti E&(A)

since a(A)c Endn A g MZ7(8), where in general the matrix algebra M,,(G) satisfies the Standard Identity S,, by the Amitsur-Levitzki theorem

209

STRONGLYPRIMEQUADRATICJORDANALGEBRAS

[25]. Hence (16.8) will follow from (16.9) and the case n = 54 of the general fact (16.10) If J is nondegenerate and satisfies S,(U,,,-lU,, U,, = 0 for all 2, x,, .... x, in J U,m-2 u,, u;2, ...) U;m-nU.rnUU;n) and all m > n, then all nilpotent elements of J have index n implies z”- ’ = 0, so we can work our way down to zn = 0. The way we will prove z”- ’ vanishes is to prove that it lies in Y(J). From zm = 0 the operators T,= U2m~~U,,U, form a uniqueness sequence: there is only one way to line the operators getting zero since TiTj = 0 if i >j (it equals up without U, Uz, where zp = 0 for all p > m by nondegeneracy once U*m-lUx,Up+l-, yrn ri = 0), thus in S,( T,, .... T,) = 0 only the term T, ... T,, survives to be set equal to zero, 0 = S,( T, , .... T,) Up-t-n= Uzm~~U,,U,m-~U,2~~~ Uzm-lU,~Uv,,-,. But this forces w =zmpl into Y(J) by (16.11)

U,U,U,....

U,U,U,.=O=-WEP(J).

(Working in the nondegenerate algebra J/Z(J), a nonzero element w # 0 would generate arbitrarily long strings y, = U,, U,, U,, . . . U, U,,_, U,,,xr # 0 by induction: w#O*U,#O+some y,=U,.x,#O, and once y,#O has been constructed there is a nonzero yr+ , in U,;J= U, U,, U, . . . un’ K, uwux,uwuxr-,u,~~~~ uw K, uw J = U, U,, U, . . . U,, ~, U, U,, U, J. ) Thus zm-’ EY(J)=O by (16.11) and nondegeneracy, and (16.10) is established. 1

17. RADICALS OF P.I. ALGEBRAS In our treatment of Clifford identities it would have been convenient at several places to be able to work with semiprimitive algebras instead of nondegenerate ones. Here we show that in the presence of any polynomial identity (e.g., a Clifford identity), SNil(J) reduces to 2(J), so that any nondegenerate P.I. algebra is actually strictly-nil-free and hence has a semiprimitive scalar imbedding. Recall the definition (7.2) of J satisfying a polynomial which by (7.2’) may always be assumed to be multilinear.

identity f=O,

17.1. P.I. RADICAL THEOREM [31, 361. Zf J satisfies a polynomial identity, then SNil(J) = Y(J), so J is nondegenerate ifs it is strictly-nil-free. Proof:

Since always SNil(J) 1 Z(J),

we need only prove SNil(J) c

210

MC CRIMMON

AND

ZEL’MANOV

Z(J), i.e. that SNil(J) is Y-radical. Since SNil(J) inherits the polynomial identity, we may replace J by SNil(J) and show that a strictly nil P.I. algebra is T-radical. By the usual radical surgery we may replace J by the nondegenerate J/p(J) and show that a nondegenerate strictly-nil P.I. algebra is zero. Further, by the Bounded Index Theorem 16.3 it will suffice to prove that J is strictly nil of bounded index. Strictness is no problem: to show xd= 0 for all x in J[T] (T= { tr, t2, ...}). note that J[T] inherits the same identity by strictness (7.2(i)) (or note that multilinear identities always extend), J[ T] remains nil by strict-nilness of J, and J[ T] remains nondegenerate. Thus we may replace J by J[ T] and need only show (17.2)

If J is nil and nondegenerate and satisfies a polynomial identity, then J is nil of bounded index: there is a d such that xd=O for all XEJ.

The index bound for special algebras is quite simple binatorics are quite delicate!): (17.3)

(but the com-

If J is special and satisfies a polynomial identity of degree n, then zm=O for m>(2n+l)s=-z”-“EL?(J). In particular, if J is also nondegenerate then all nilpotent elements have index <2n.

The last part follows from the special case s = 1: zm = 0 for m~2n+l*z”~‘= 0. (On the other hand, for the general non-special case it turns out to be the case s = 3 that we need.) To prove the first part, start from z”’ = 0 for m 2 (2n + 1) s and construct for any xi E JC A a uniqueness sequence zi = {z(‘- ‘Is, xi, z”~ C+ l)‘} E J: assuming the P.I. is multilinear as in (7.2’) we use the underlying associative product in A and the relation zm = 0 to see (*)

0 = zm- yqz,, = z-y =

z2, ...) z,) zns

lx1z~-2~)(z~x2z~-3~).

Zm-sX,Zm-sX2..

. . (z (n-l)SX,Zm-(n+l)S)Zns

. Zm-SXnZm-s.

(Indeed, note that each zi is surrounded on each side by at least (i - 1) s factors since m-(i+l)s>m--(n+l)s>(n-l)s>(i-1)s for m>2ns; thus in the expression (7.2’) forf” the front factor z+~ kills z2, .... z, (from zm = 0), so only the permutations with rr( 1) = 1 contribute, and z, contributes only the factor l~~z”-*~ since m - 2s 2 s as long as n > 1 (the case n = 1 is trivial, f(xl) = x1 vanishing forces J= 0, so we always assume n 2 2.) If we have argued that only rt fixing the first i terms contribute, contributing a leading term ~~-~(lx~z”-~~). . . (z(~-‘~~x~z~-(~+‘)~), then

STRONGLY

PRIME

QUADRATIC

JORDAN

ALGEBRAS

211

the last factor kills z~+~, .... z, since they are all surrounded by at least (i + 1) s factors z, so the next surviving term must be zi+ , , rr must lix i+l, and only the term (z~~x~+~z”--(~+~)~) from zi+i contributes (since m-(i+1)s+m-((i+2)s=2m-(2(i+l)+1)s~2m-(2n+1)s [i + 1
so that w = zm-’ satisfies U, U,, U, . . . U,,, U,, Uwxr+ , = 0 for all xi E J, therefore w E Y(J) by (16.11) as desired. The situation for general quadratic Jordan algebras is more involved, for there seems to be no very useful “canonical form” for quadratic Jordan polynomial identities. We will use the Absorber Trick introduced in [29] to pass from an element to an inner ideal to a special algebra, where we can use the above bound. If B is any inner ideal in J there are quadratic and hear absorbers q(B) c I(B) c B which are ideals of B and inner ideals of J such that (17.4)

(i)

B/l(B) is special

(ii)

1(B)3cq(B)

(iii)

{ Uj+

Vj,j} q(B) c B

(cf. [2, 7.2.6, 7.2.131 or [9] for the quadratic case). We also need a slight generalization of Inner Transfer (16.4). Let us recall the recursive construction of Y(J) as the limit of ideals Yi(J) for ordinals L defined by

Z,(J) = 0, Z(J)= The Structural (17.5)

Transfer

Y;+,(J)

= span of Iz 1 U,Jc

t-j yv:(J)

5$(J)},

(A a limit ordinal).

lr
Principle

[ 31, Lemma 15; 191 says

If T: J, + J2 is a structural transformation between Jordan algebras ( U$?/x, = TUil)T* f or some linear T*: J2 + J1 and all XE J,), then for all ordinals T(5$(J,)) c L&(J,).

We claim that for general algebras we have the following improvement on (17.2) (three times weaker than (17.3) for special algebras): 607/69/2-b

212 (17.6)

MC CRIMMON

AND

ZEL’MANOV

If J satisfies a polynomial identity of degree n, then = zm=O for m> 6n+2 implies z”-l ET(J); in particular, if J is also nondegenerate then all nilpotent elements have index .< 6n + 2.

p-1

To apply the Absorber Trick we introduce the principal inner ideal B=@z+

U:.?.

This and its special image B = B/I(B) (by (17.4i)) inherit the polynomial identity of degree n, so for m > 6n + 3 = (2n + 1) s for s = 3 we have zm = 0 in J * Z in i? has Z” = 0 (here it is important that z lie in B so B = U,j will not work) * ZM--S E 9(B) = c (by (17.3) for the special algebra B) * m-3 E C, for C the smallest ideal of B containing I(B) with B/C nondegenerate. Now B/q(B) need not be special, but we still can replace I(B) by q(B): by (17.4ii), I(B) is nilpotent modulo .q( B), so the smallest ideal D of B containing q(B) with B/D nondegenerate automatically contains I(B), so q(B) and j(B) have the same Y-closure D = C in B, and

We now apply the Structural Transfer (17.5) to J1 = B/q(B), J, = J, T= UP-1 U, U=, T* = Uz U, Uzm-1 for arbitrary x E J. Here T and T* will form a structural pair by (0.3) as soon as they are well defined. T* maps J2 into U, Jc B, which then is canonically projected onto J1. T certainly maps B into J, and it induces a map on B/q(B) since necessarily T( q( B)) = 0: by its absorptive nature (17.4iii) with respect to J and its ideal nature with respect to B we have U,m-l Ui, U,q( B) c UP-, U,q(B) (here again we need ZEB)~U,,-,B=U,~~,(~Z+U,~)=~Z~~-‘+U~~=O by our two hypotheses on the nilpotence of z. Thus the hypotheses of (17.5) are met, and we conclude that Y(J) = P’(J2) 2 T(sP(JI)) = T(Y(B/q(B))) = T(D/q(B)) contains T(ZmP3) (see above) = UP-l UxUz(zmP3) = Uzm-, Uz:Zn’-l for all x. But in general (17.7)

U,U,WELZ’(J)+WEY(J).

(This is easy for linear algebras. For quadratic algebras we pass to the nondegenerate algebra J/Z(J) and prove U,U,w = 0 *w = 0. Now we continue to have U, UP U,p = 0 for all p E J[t] since J[t] remains nondegenerate and Uu~w~u~P~u~w~P= U,U,U, UP U,U,U, = U U(w,U(P,wUP U, = 0 (note U, U,w = 0 * U, U,,,w = 0 =. U, U,w = 0 in all scalar extensions of J). Identifying coefficients of t in U, UP U,p = 0 for p =x + ty shows U, U, U, y + U, U,>, U,x = 0, so Uu(,+ y = U, U, U, y =

STRONGLY PRIME QUADRATIC JORDAN ALGEBRAS

213

w, w, y } (by (0.24)) E U, U,,w = 0, so all U,x -K{x, Gw}= -K$L are trivial and hence zero by nondegeneracy, so w is trivial and hence zero.) Thus w = zm- ’ lies in 9’(J) as required by (17.6). 1

INDEX OF DEFINITIONS

We refer to the place where the term is first introduced in the text, in terms of the nearest labelled formula (n+ means after, n- means before the formula with label n). A

1.4+ 13.1313.4, 13.6 0.10 15.2 16.10.8’ 0.13’ 0.8’ 5.1 0.16 3.1 13.6+

absorber AbsJ~,(K, I) of an ideal abstemious algebra adic family {F.} albert algebra albert form albert identities Alb(X) ample hermitian algebra H,,(A, *) ample outer ideal I * J ample subspace of hermitian elements H,(A, *) anisotropic quadratic form annihilator (two-sided Ann,(l), inner Inann,( anti-hermitian type associative n-tads A,(x,, . ... x,) B

0.15+ 4.1 5.1 -

Baer radical B(J) ( = semiprime radical) big extension bilinear radical Bilrad(Q) of a quadratic form

8.2 4.3+

center C(J) of a linear algebra central closure central algebra (F(J) = @) central algebraic closure centrally algebraically closed centroid F(J) Clifford Jordan algebra J(Q, 1)

C

4.3 4.3 4.2 0.9 6.115.2 7.2+ 7.2 + 7.2+ 0.25-0.27 13.6 12.7 0.14-

Clifford form Clifford identity clifford polynomial Clifford type commutator [x, ~1 compatible with tetrads, pentads core K” of an outer ideal core fi of a subspace

214

MC CRIMMON AND ZEL’MANOV D

5.1 5.10.12 0.14 0.12+ 0.12+ 0.15+ 7.11-

defect of a quadratic form defective quadratic form derivation (inner D,, in 0.27) derived ideals I’“) direct product fl J, direct sum J, H J, division algebra doubly interconnected idempotents E

2.3 0.9 + 2.5. 0.9+

eating (n-tads (12.1), imbedded n-tads 13.1, S-n-tads 12.9) envelope (+-envelope, tight envelope, *-tight envelope) envelope absorption exceptional ( = not special)

13.1,

F 1.116.1 0.4 l.l0.6 6.1+ 1.1

free associative algebra a(X) free Jordan algebra FJ(X) free scalar extension Jo free special Jordan algebra FSJ(X) = /(A’) free unital hull j form of f (J having 7 as unital-scalar extension) formal ideal in j(X) H

10.2+ 10.1, 10.2 13.8+ 0.8 3.4 3.1 1.2+ 1.2 1.2+ 1.3 3.40.11 2.8 +

Hall identity Hall-Zel’manov identity hearty n-tad-eater (space HE,, ideal H,) hermitian algebra H(A, *) (ample hermitian H,(A, *) (0.8’)) hermitian form hermitian type hermitian polynomial hermitian ideal X(X) in d(X) X-part X(J) of J hereditary hypotheses hereditarily semiprime homomorphism homotope-invariant I

0.13+ 1.1 2.4 0.9 + 0.9+ 13.6+ 13.1

ideal Cl (inner 4, outer a, ample outer 4 ,* 4 ) ideal (formal) ideal absorption i-exceptional ( = not i-special) i-special ( = homomorphic image of special) imbedded n-tads imbedded n-tad-eaters (space IE,, ideal I,)

operator eats n-tads

STRONGLY

16.4 0.21 7.110.15+ 6.1 +

PRIME

QUADRATIC

JORDAN

ALGEBRAS

inner transfer principle inner derivation D,, interconnected idempotents (doubly, orthogonal) invertible element isotope Ji”’ J

0.15+

Jacobson ( = semiprimitive) radical Rad(J) L

6.2 17.4 0.6+ 2.7 7.3 15.4+

large Clifford form linear absorber I(B) of an inner ideal B linear Jordan algebra (+ E @) linearization-invariant ideal, closure local algebra of J locally nilpotent ideal M

0.33 1.5+

Macdonald’s principle Martindale *-algebra of quotients Q(A) N

3.1 3.2 0.15+ 0.15+ 0.15+ 0.15+ 0.15+ 5.1 14.7 1.2+ 12.1-

natural specializing ideal SP,,,~(J) natural specialization Y,,r,“(J) nil ideal (strictly nil, nil radical Nil(J), strictly-nil radical SNil(J)) nil-free algebra nilpotent element 2” = 0 nondegenerate algebra nondegenerate (lower) radical nondegenerate quadratic form norm n(a) = aa* in a *-algebra n-tad {xi . ..x.} n-tad-eating ideal T, 0

13.6+ 7.11-

ordinary n-tads T,(x) = {xi orthogonal idempotents

x,} P

1.2+ 12.5 I 7.117.2 7.2+ 15.2 0.15+ 0.15+ 17.6+ 0.14-

pentad {x~x~x~.LA~ Peirce spaces J, polynomial identity (multilinear 7.2’, satisfies 7.2) PI. algebra P.I. degree 13 powers x” of an element prime algebra principal inner ideal product of ideals (I,, I,

215

216

MC CRIMMON AND ZEL’MANOV

Q 0.1-0.3 5.1 17.4 5.1 0.13+

quadratic Jordan algebra (unital; non-unital 0,5,0.6) quadratic form Q quadratic absorber q(B) of an inner ideal B (quadratic) radical Rad(Q) of a quadratic form quotient (factor) algebra J/f

R 16.15.1

radical identities radical Rad(Q) of a quadratic form Q (bilinear radical Bilrad(Q)) S

8.3 0.4 0.15+ 0.15+ 0.33 0.15+ 6.2 3.2+ 0.15+ 0.7 + 3.2+ 16.17.6,7.6’ t6.9 + l.l0.3+ 0.15+ 0.15+ 0.15+ 17.5 0.140.12+

scalar element scalar extension (free Jn; general QJ 0.19+ ) semiprime semiprimitive (semisimple), radical Rad(J) Shirshov’s principle simple algebra small clilford algebra, form smallest specializing ideal Sp( J) solvable ideal I@) = 0 special Jordan algebra J c A + speciahzer ( = smallest specializing ideal) s-identities Y(X) standard clifford identities standard identity S, standard (main) involution * on a(X) strict identity fz 0 strictly-nil ideal strongly prime = prime + nondegenerate strongly semiprime = nondegenerate structural transfer principle subalgebra G(X) generated by X subdirect product (sum) z n Ji T

1.2+ 12.5 I 2.7 0.16 0.9 + 0.16 0.18+ 6.2 0.10 14.7

tetrad {xIx2xXxq} T-ideal, T-closure tight cover tight envelope :-tight envelope tight unital-scalar extension tiny chfford algebra, form Tits Construction (First 0.101, Second 0.1011) trace tr(a) = c + u* in a *-algebra

STRONGLYPRIMEQUADRATICJORDANALGEBRAS

217

U 16.10+ 17.3+ I 13.4, 13.6 0.6 13.4 0.19+ 2.6 16.1

uniqueness sequence unital adic family {F,} unital hull j (free) unital Jordan-alternating properties unital-scalar extension J= szij universal absorption universal Jordan algebra % (free on infmite set of generators) V

7.3

vanishes locally, globally 2

14.3

Zel’manov polynomial

INDEX OF NOTATIONS

We refer to the place where the symbol is first introduced in the text, in terms of the nearest labelled formula (n + means after, n- before the formula labelled n). A

l.l-, 1.4 2.8 + 0.7 13.72.1 16.10.15

a, a(x) aId Af A,@,, ...> 4 AbsJ,.,(K 4

a, alb(x) Am(I)

free associative algebra on a set X homotope with u yb = ayb associative algebra made quadratic Jordan associative n-tad absorber in 7 of K into I in J albert identities in variables A’ two-sided annihilator of ideal I in J B

5.1 0.15+ 3.3+

Bilrad(Q)

o(J) BAJX sAJ> 0 char(@)

8.2

C’(J)

bilinear radical of quadratic form Q Baer (semiprime) radical of J Ith term in construction of Baer radical of J or of Jmod I

characteristic (char(@) =p op@ = 0) center of linear algebra J D

0.27 14.1 0.14 4.1

Dx,,, D 92 D(I), D”(I) dim, J

inner derivation ad[x, ~1 subalgebra of d(X) generated by D.&F) derived ideal of ideal I dimension of J over a geld @

218

MC CRIMMON

AND

ZEL’MANOV

E 12.1+

EmEn(X)

space of n-tad-eaters

7.1 13.813.713.7 13.916.1 16.1l.l-

fE0 9

FJ(X)

f vanishes strictly a collection of adic-families an adic family imbedded adic family induced adic family free Jordan algebra on a set X

FSJ(X)

free special Jordan algebra on a set X

F

;z:w, FT

H 0.8

WA, *)

0.8’ 7.1

H&A *) H, = H(M,,(@))

0.14 13.1+ 13.1+ 0.13+ 0.15’ 3.1 1.2+

I’“’ I& In 4(X), 9”(X)

hermitian (= symmetric = selfadjoint) elements of associative algebra A under involution * ample subspace of hermitian elements hermitian n x n matrices over 0 I

Inann,

i

y(J)

n th derived ideal D”(I) space of imbedded-n-tad-eaters ideal of imbedded-n-tad-eaters ideal in J, A generated by subset X inner annihilator of ideal I in J S n Y,(1) associative closure of ideal I in S ideal of values taken on in J by a formal ideal f(X) J

l.l-

93 d(X)

quadratic Jordan algebra free scalar extension scalar polynomial extension in indeterminates T scalar polynomial extension in r indeterminates scalar rational extension (scalar denominators) free unital hull factor algebra of J by ideal I Peirce space relative to orthogonal idempotents central closure u-isotope of J Tits First Construction Tits Second Construction Jordan algebra of admissible cubic form N Jordan algebra of quadratic form Q with basepoint (Q(l)= 1) free special Jordan algebra FSJ(X) on set A’

12.7

?P

core of a subspace (maximal ideal inside it)

0.1 0.4 0.4+ 4.1 4.1 0.6 0.13+ 7.11 4.1+ 6.1+ + 0.101 0.1011 0.10 0.9

J

Jfl JCTI JCt19.... 171 J(T) j J/I J-q J J’“’

J(A ~1 J(A *, u, ~1 J(N, 1) J(Q>1)

K

1

STRONGLY PRIME QUADRATIC JORDAN ALGEBRAS

219

L 2.6+ 0.15+ 11.4 17.4+

I(a) g(J) l(B) Z,(J)

length of element a lower or degenerate radical of J linear absorber of inner ideal B Ith stage in construction of Y(J) M

0.8 14.1

M,(D) A,

x n matrices over D outer multiplication by J n

N

0.15

Nil(J)

7.9d-

Nd J)

nil radical of J alternative nucleus of J P

14.1 14.1

P12.P16,P23,P48

P8

hearty tetrad-eaters possible tetrad-eater

Q 0.15+ 1.s+ 14.2 14.3 17.4

q.i.

Q(A), Q,(A) 948

Q&W q(B)

quasi-invertible (1 - x is invertible) Martindale *-algebra of quotients relative to filter 5 of *-ideals hearty pentad-eater of degree 48 (a Zel’manov polynomial) T-ideal in j(X) generated by q48 quadratic absorber of inner ideal B R

0.15 5.1

Rad( J)

Ra4Q)

Jacobson ( = semipripitive) radical of J radical of quadratic form Q s

0.15 3.1 3.2 3.2+ 16.119.9 +

SPLSAJ) YI.S.A SP(J) Y(X) S”

strictly-nil radical of J specializer of J with given data natural specialization of J with given data specializer of J (largest special image) ideal of s-identities in FSJ(X) standard identity of degree n

tr(a) MA 1 @!4:(4) T,(X) TAX,, ...>x,) pyx,, .... x,) T(x)

trace a + a* of element a set of all traces of A generalized traces n-tad-eating ideal ordinary n-tad imbedded n-tad trace Q(x, 1) in a Clifford algebra

SNiI( J)

T 1.915.5+ 2.5i 12.1 12.1 13.75.1

220

MC CRIMMON

AND

ZEL’MANOV

U 0.1 0.1 16.1 14.3

u, u I..” *z U,I

U operator (analogue of z --) xzx) polarization Cr,,, - uX - lJY (analogue z -+ xzy + yzx) universal free algebra FJ(X) on infinite set X symmetric matrix units E, + E,l V

0.1 0.6+ 1.2 1.2+ 2.1 2.7 3.3+ + 7.3 13.8+ 13.8+

V *.v VX S(X)

*o(J) 2 X’ HI WY,, ‘1.3Y,) HE,, Jf-40’) H,

operator z + { xyz} = U,,, y V, = V,., circle operation V,z = x 0 z hermitian ideal in universal j(X) values of hermitian ideal on J T-closure of X(X) linearization-closure of S(X) anti-hermitian part hereditary hypotheses on y,, .... y, space of hearty n-tad-eaters ideal of hearty n-tad-eaters W

0.26

W X..”

operator (analogue z + {xyzxy }) Z

0.15+ Greek

Mathematical

0.13+ 0.1 1.2+

n th power of an element

r(J)

centroid of J field of fractions of domain

Letters

4.2 4.4 1.32.9 0.1 0.140.19+ 6.1

0.12+ 0.12+ 0.12+ 0.12 0.12+

z”, x”

r

directional derivative at a of p in the direction b @ @(X) s Symbols .-.E

a

iJ, z n J,

-cl 4 -a + 4 IXYZJ ix* . . ..%I

ring of scalars (commutative associative with 1) subalgebra of J generated by subset A’ unital-scalar extension field of square roots of 0 in characteristic 2 equals by definition congruent to (module something understood) algebra direct sum module direct sum direct product of the Ji a subdirect product of the J, ideal inner ideal outer ideal ample outer ideal *-ideal Jordan triple product VX,Zy (analogue 3-tad) n-tad x,x2 . ..x.+x,...x,x,

STRONGLY PRIMEQUADRATICJORDAN 0.21' 0.6+ 0.6+ 0.25 ff.

cx.Y, 21” X.Y XOY

CX?Yl

221

ALGEBRAS

circle associator (xoy)oz--xo(yoz) dot product (analogue 1/2(xy + yx)) circle product (analogue xy + yx) commutator of x, y (does not exist as element, only in operators)

CX>Y12

0.25 0.26 0.21

UCX..“l

CC~~Yl.~l

Dx,.&)

ACKNOWLEDGMENTS The authors thank the organizers of the 1982 Conference on Radicals in Eger, and the 1983 International Congress of Mathematicians in Warsaw, for providing the opportunity for this joint work. The first author was partially supported by the National Science Foundation under Grant MCS-80-02319, and also thanks the Technical University of Munich for its hospitality during the final preparation of the paper.

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