- Email: [email protected]
Hopf algebras and derivations
JOL’RNAL
OF ALGEBRA
64, 2 18-229
(1980)
Hopf
Algebras
and Derivations
Gl!NNAK
SJaDIN
Department of Mothemotics, University of Stockholm, Box 6701, S-113 85 Stockholm 23, Sweden Communicated
by D. Buchsbaum
Received May 9, 1978
Let A be a connected Hopf algebra of finite type over a field k. Then it is proved in Andre [I] that A* = UL, the universal enveloping algebra of a Lie algebra L, if and only if -4 is a Hopf algebra with divided powers. In [I] the concept of Lie coalgebras is introduced and the structure theorem above is proved and stated in its dual form. The main purpose of this paper is to prove this result directly. It is possible to give a short proof of the “if” part using a technique with derivations. This is done in Theorem 1. At the same time we prove that .4 is free as an algebra with divided powers, which extends the theorem of I,eray (cf. !Uilnor-I\‘Ioore [3, p. 2521) to arbitrary characteristics. In [I] it is supposed that there are divided powers in each dimension if char(k) : 2. Since the author is interested in applications to the Ext and Tor algebras of a local ring we do not use this definition for char(k) = 2. Instead we arc forced to introduce a slightly modified form of Lie algebras, which we call adjusted Lit algebras. If char(k) ;’ 2 th en an adjusted I,ie algebra is simply an ordinary I,ie algebra. We conclude the paper by getting the Poincare-Birkhoff-Witt theorem, for adjusted Lie algebras, as a corollary to the fact that the dual of the enveloping algebra is a Hopf algebra with divided powers.
%OTATIOSS
AND
CONVENTIONS
In the sequel k will denote an arbitrary field. All vector spaces, tensor products and direct sums are over K. We shall use the following symbols, definitions and conventions. I.
The sign # in a diagram
2.
In expressions
means that the diagram
for signs depending
218 0021-8693;80,050218-12$02.00/O Copyright 0 1980 by Academic Press, Inc. All rights of reproduction in any form reserved.
commutes.
on scariables given some degrees we
HOPF
ALGEBRAS
AND
DERIVATIONS
use the elements themselves to express the degrees, e.g., we write instead
of
3.
(-
219 (- I)a’b
])dWa).deW).
The letter N denotes the set of natural numbers or a finite part of it.
4. If X = @i>O Xi is a graded vector space of finite type then the Hilbert series H, is the formal power series defined by H,(z) = &,, (dim,X&?. If X,, = 0 we let fix denote the formal power series defined by $-1 Hx(z)
Obviously
=
g
“,I’-
Hx g,r = H, * H, and fixc,’
dlm,x,,-,
$ilm,x,.
= fix * fir .
5. For a graded vector space X we put X+ ~7 oi>l Xri and X- = Oi>Ox*i+~~
6. Suppose that X is a graded vector space. Then “x E X” always means that x is a homogeneous element of X, a basis of X means a homogeneous basis of X, etc. 7. If X, Y are graded vector spaces then X @ Y is a graded vector space with (X @ Y),, = @f+j,n Xi @ Yj and Hom,(X, Y) is a graded vector space with Hom,(X, Y)n = niPjcn Homk(Xi , Yj) = the set of k-linear maps f: X + Y such that f(Xi) C Yi-n . 8. As for signs we use a strict sign convention. This means that as soon as we define anything by means of a formula we should multiply by a sign related to the re-ordering of elements. The meaning of this principle is explained in the sequel. Let x = (x1 ,..., x,,J be a sequence of some elements, each of which is given a degree. Let (I E S(m) and put u(x) := (x0-r(r) ,..., x,-I(,,,)). Note that u(x) is obtained from x by moving the element in the ith place to the o(i)th place. It follows that (UT)(X) = u(T(x)) for (I, T E S(m). We put f(u, x) = (~)~~OlJ~“l’*~ . It is obvious that ~(7, u(x)) * ~(u, x) = l (TU, x), which shows that any two sequences of re-orderings of x ending up in the same final ordering induces the same sign. This may be expressed by saying that all diagrams are sign-commutative. Then, to illustrate our principle, if f E Hom,(X’, X),g E Hom,(Y, Y’) we define Hom,(f, g): Horn&X, Y)-+ Hom,(X’, Y’) by Horn&f, g)(h) = (- ly’(g+*)g 0 h 0f = e(u(f, g, h))g 0 h of, where (u-l(l), u-l(2), u-l(3)) = (2, 3, 1). It follows, by the above-mentioned principle of signcommutativity, that Hom,( f',g') 0 H omk(f,g) : (-l)f.(f’+g’) Hom,(f o f', g’ 0g) if f' E HomJX”, X’), g’ E Horn,. Y’, I;“). Kate that, in accordance with the principle, we define the biduality map h: X + X**, where X* = Hom,(X, k), by the formula h(x)(f) = (- l)““f(x). Similar considerations apply to other maps, e.g., 5: X* @ X* --, (X @ X)*. 9. Let X be a graded vector space. Then Hom,(X, X) is a Z-graded algebra with composition as product. If X,, = k then we regard X* as imbedded in Hom,(X, X) via k --f X.
220
GL’NNAR
SJijDIN
10. The commutator [ , ] in a graded algebra (meaning associative algebra) A is defined by [a, b] == a . b - (- l)a*bb . a. We say that A is commutative if [ , ] = 0 and that it is strictly commutative if furthermore a2 = 0 for a E A- . 11. A Y-algebra A is a strictly commutative graded algebra with divided powers of elements in A., , satisfying the condition of [I, p. 191. Note that we have the same definition if char(k) = 2, contrary to the case in [I]. We denote the exterior powers of x E A+ by x(n). For any .t E ,4 let x(O) == 1 and x(l) := x. If A is augmented, e.g., if A is connected, and IA is the augmentation ideal then we put DA Z*A $ &>s ky,(A+), where m(.r) = ,Y(“). If X is a graded vector space then Z’X denotes the free r-algebra on X. Note that H, = ZZr . 12. A Hopf r-algebra A is a Hopf algebra, which is also a r-algebra, such that d: A - ‘4 @ A is a homomorphism of Z-algebras. Using the universal property of the functor Z’ we see that if X is a graded vector space then there is a unique Hopf r-algebra structure on Z’X such that XC PI’X (actually there is then equality, which is an easy consequence, e.g., of Theorem 1). The category of connected Hopf r-algebras (of finite type) will be denoted by HOPFr (HOPF=‘). 13. Let X be a graded vector space. Then algebra on X (the tensor algebra). 14. An adjusted tural maps
Lie algebra I, is a graded
I, @L -‘L’L- -5 such that there is a monomorphism is a graded algebra, satisfying
7’(X)
is the free noncommutative
vector space L with two struc-
L of degree 0, L doubling
the degree
i: L -+ A of graded
vector spaces, where
A
and
L- --K-e
L
d # A- a~
Ii .4
If char(k) # 2 then an adjusted Lie algebra is just an ordinary Lie algebra. With the usual commutator and square any graded algebra may be considered as an adjusted Lit algebra. -4s in the usual case there is an enveloping algebra
HoPF
ALGEBRAS
AND
221
DERIVATIOSS
WL = TL/cII, where 2I is the ideal generated by elements [x, y] - i,x, y) for x, y EL and x2 - K(x) for x EL- . Note that L + WL is a monomorphism of adjusted Lie algebras. If X, Y are adjusted Lie algebras then so is X @ Y with <(x, y), (x’, y’)) = ((x, x’), (y, y’)) and K(x, y) = (KX, KY) and it follows by the universal property of W, that W(X @ Y) -+- WX @ WY, where (x, y) .. + x @ 1 - 1 6~ y for x E X, y E Y. It also follows that WL is a Hopf algebra with diagonal A: WL -Pd W(L @L) -+* WL @ WL, where d(x) r= (x, x) so that L C PWL. ,4n adjusted Lie algebra is said to be connected if L, = 0 and trivial if ,’ , ) and K are 0. The category of connected adjusted Lie algebras of finite type is denoted by LIE’. Sote that if L is trivial then WL is just the free strictly commutative algebra on the graded vector space L so that if L E LIE’ is trivial then H,, = Z^i, . If L is an adjusted Lie algebra then L* denotes the trivial adjusted Lie algebra with the same underlying graded vector space as I,.
DERIVATIONS
AND
THE
STRUCTURE
DEFISITIOK. Let A be a Z-algebra. derivation on A iff(xy) = f(x) . y -I- (The set of derivations on A is denoted In the following Hom,(A, A) is given
LEMMA
Hom,(A,
1.
OF ANDRE
Then f~ Hom,(A, A) is said to be a 1)““~ . f(y) andf(+)) = f(x) . xc” l). by Der A. an algebra structure via composition.
Let A he a r-algebra. Then Der A is a subadjusted Lie algebraof
A).
f, g
Proof.
Let
(a)
[f,g]
(b)
.fz E Der A if deg(
E Der A. We have to prove that
=fo.q--(-l)“~~cf~Der.4,
f)
(a) By direct computation [f, g](-y). Let x E A, . Then [f,gp'
THEOREMS
= [f,g](x)
is odd. we get v, g](xy)
*xc" -1) $ ((-l)"'"'"'g+~
and the fact that A is commutative Thus [f, g] E Der A.
= [f, g](x) . y + (- l)(~+O)‘s.~ *
.fx-(-])/'"7"'"4f~
.gx) . X(w-2)
and deg(x) is even makes the last term vanish.
(b) \Ve have f z(xy) = f'(x) * y - (( - 1)‘*(f+2) last term vanishes since deg( f) is odd.
-b (-
l)““)&x
* f y and the
Let x E A,. Thenf“x(n) zz f'(x) * ~('1-1) + (-l)"(fi")fx . fx * .~['+2) and fx .Jv = 0 since deg(fx) is odd and A is strictly commutative. It follows that f 2 E Der A and hence that Der A is an adjusted Lie algebra. The construction in the following lemma is basically the same one as in Gullikscn-Ixvin [2, p. 1091.
222
CUKNAR
LEMMA
SJGDIK
2. Let A E Hopfr and let for f E A* = Hom,(A, v(f) = j=E-4 -%
ABA
K)
‘@‘+A@k=A
Then, (a) A* -+” Hom,(A, A) is an algebra-ho momorphism (note that A* is always an algebra but not necessarily a Hopf algebra, unless A is of finite type); lb) (c)
= f; E Der A -f(DA)
v(f)/Ade,tf)
v(f)
=. 0;
(d) PA* = {f / f(DA) = 0} is a subadjusted Lie algebra of A* p?4* --ty Der A is a monomorphism of tijusted Lie algebras. Proof.
(a)
and
Let f, g E A* and consider the diagram A
LABA
1
A
A@A
#
&A
*@‘+A@A@A
We have 4g) o v(f)
= ((1 Og) o A) o ((1 of)
QA) = i +*
and
whence dg) 0 v( f ) = v(g * f ). (b)
This follows from the fact that d has a co-unit.
(c)
Suppose that f (DA) = 0. Then for x, y E IA
3(-v) = (1 c3f)4XY) = (1 Of)+)
=3(x>* y
= (1 Of)(&)
. (y 0 I) + (x 0 1>4YN
* (Y 0 1) + (-l)““(X + (-1px
0 I)((1 Of)4YN
.3(Y).
Suppose that x E A+ and let d(x) = x @ 1 + 1 @X + xs
c (x(Q) @ I)(1 @ x’““)(x; @x;)‘““’ **. (xi @ x:)‘“J. s,+...+s,=n.si*
HOPF
ALGEBRAS
AND
223
DERIVATIONS
But, (xi @ xp
=
if xi, xl are of odd degree and s > 2 O sy 0 .@) if x:, x; are of even degree
so it is sufficient to consider those terms for which si y= 0, 1 for 2 < follows that f,(n) :.= f(x) . Jn-1) .+ c
f($)
. J-1)
i <
Y.
It
. x;
3&q
==(lOf)(xOI+lOx+
c
x: @ A$ * (X-1)
@ 1) =i(x)
* Xc+1).
1
3(i
Now, suppose that v(f) E Der A. Then we have to show that f(DA) = 0. Let x, y E IA be such that xy E Adee . Then f(xy) = f(zy) = 3(x) * y + (--l)l’o~ .3(y) = 0 since deg(x), deg(y) < deg@). If x E A+ and rz 3 2 are such that x(n) E Adeg(,) then J+) = ffrcn) =3(‘(x) * @-I) == 0 by the same reason. Thusf(DA) = 0. (d) Let f,g E PA*. Then r$J g] = [df), v(g)] E Der A and hence u, g] E pAA* according to (c). Similarly, iff E PA* is of odd degree thenf* E PA*. Furthermore, v(f) = 0 implies that f = r( f )/Adeg(,) = 0, which shows that Y: PA* + Der A is a monomorphism of adjusted Lie algebras. I. (a) Let A E HOPFr. Then any section of IA -+ IA/DA induces ismorphismr(IA/DA) -+ a A of F-algebras. (b) If A E HOPFrf then PA* CA* inducesan ismorphismWfrA* -+c A* of Hopf algebrasand HA == I?,,,,, . THEOREM
an
Proof. Let A E HOPF r’. By the universal property of W we get a homomorphism/3 of algebrasasabove. Since PA* C PA* this is alsoa homomorphism of Hopf algebras.Let IA = DA @ X as a graded vector spaceand let {xJ~~,.,, where IV denotesthe set of positive integers (or a finite part of it), be a homogeneousbasis in X such that deg(x,) < deg(xj) implies that i < j. Consider the set ‘3 of sequencesY = (ri)isN of nonnegative integers such that ri = 0 for i > n(y) and such that ri = 0 or 1 if deg(xi) is odd. Order 2I by letting Y > Y’ if the last nonvanishing ri - r: > 0. Let fj E A* be given by -f;(p) =,fv and f,(DA @ k) = 0. For Y E 2I with rr = n(r) we put .@) = x,” a.0x1 ’ and f’=fi ***f 2. We have u(f’) = v(fJ’l o ..* o v(f,,>‘ll where the exponents now are with respectto composition. It follows that
v(f’) d”) = 0,
I
if if
Y>Y’
Y=
Y’
and in particular the sameis true for f rx(r’). Now, supposethat xreB &..z(r) = 0, where 23 is a finite nonempty subset of ‘3 and X, # 0 for I E!B. Let s be the
223
CUNNAR
SJijDIN
largest element in 23. Then 0 -= f Yxres /\,.T(~) = h, , which is a contradiction. Hence the x(‘): s arc linearly independent and since, by induction on the degree, it is seen that they generate A it follows that they constitute a basis for A. In particular XC A induces an isomorphism of r-algebras rX - A, which proves (a) when A is of finite type. In the general case TX --f A is an epimorphism by the same argument as above. To prove that it is a monomorphism it is enough to notice that any finite-dimensional vector space I/C X is contained in a sub-Hopf r-algebra of finite type. Another possibility would be to repeat the former argument but now using a well-ordered basis. Kow, let us turn to the proof of (b). According to the formula above {f r},Ell is expressed in the dual basis of {x(‘)} rest by a triangular matrix with 1 : s in the diagonal and hence (f r},es is a basis for A. In particular {fi}ieN generates A* as an algebra. Since fi E PA* this implies that fl is an epimorphism. By (a) HA. = HA : HrcIAiDAj = It?,,, ,DA = - Z?JS~,(this shows the formula in (b)) and hence it will suffice to show that HH.pA. ,< Z?p,. . We prove that H,, ,< flL for anyL E LIE’. Filter WL by F, WL = the elements of WL that may be written as polynomials of degree
Let PI,,,,, be the set of sequences (E -= (C, ,..., C,,), where
(a) Ci is a subsequence, possibly empty, of (I,..., m). (b) The underlying subsets of the Ci : s are disjoint and their union is {I,..., m}. Let f%,, be the set of sets D =- {C, ,..., C,}, where the Ci : s satisfies (a) and (b) above. Note that we have a surjective mapping PI,., --f 9,,*, given by 0. = (C, ,...) C,) + E = {C, ,...) C,}. As usual, if T E S(n), we put ~((5) = Then oe E S(m) is given by (C’(1) ,*.a, C,-,(,j). Let Cf = (rp+r).,r ,..., yvti)). u$(j) I= rj . Let A be a commutative algebra and let x == (x1 ,..., x,,,), where xi E A. Then, if C 7. (il ,..., is) is a subsequence of (I,..., m), we put x, = xi, ... xi8 (=:I if C is empty). In particular xc1,..., m) = xr ..’ x,, . For 0. E ‘& .n we put cc2 = E(U~ , x). Ixtf, )...(f,?nE Hom,(A, A) and let y E A+ . Then, for (5 -= (C, ,..., C,) E ‘?I,,,, , let f&) : - cE’fc,(y) ... . fc,(y). Using the fact that y has even degree and A is commutative we get f,(o)(y) = fa(-v) for T E s(n). Thus we may define f&v), for a c %,,., , by the formula f%(y) -=f&(T), whcrc 0. E 91,,,, is such that c - II).
HOPF
ALGEBRAS
Let A E HOPFr
AND
225
DERIVATIOXS
and let x1 ,..., x,, E PA*, y E PA”,
LEMMA
3.
I’rooj.
It suffices to show that
y E A+ . Then
where x =_ (X r ,..., .v,,,), and since the Si : s are derivations this follows by induction on m. Now, dually, let A be an algebra of finite type and letj E AT and x1 ,..., x,,, E A. Let, for 0: E PI,n., , j(xe) = g~j(xc-) . ..**j(~c.). Then, since im(j) CK, k is commutative and j has even degree, we get j(+tc,) = j(xc). It follows that we may define j(+) for 3 =- c E d,,, by putting j(xn) :.: j(~). Inspired by Lemma 3 we prove
l
LEMMA wL,T.,,,,J,
4. Let I, E LIEJ and let f E WLZ . Then there is a unique j[“l which satisfies f [nJx,... x,,,= xFEs,., j(xx) for x1 ,..., .Y,~EL.
E
Proof. Obviously jl”l is unique and jtnl E WZ4z.dey(J, if it exists. Define jcrL) E (TL)” by j(n)~i ... x,,, = &.a,., j(x,), where Xi EL, xl .*. X, is formed in TL and the x a : s are formed with respect to the product in WL. It will then be sufficient to show that the ideal, generated by elements of type [x, y] - (x, y) with X, y EL or x2 - K(X) with .v EL , is mapped into 0 by j(“). Thus we have to show that (a)
j(R).~,
“’
Xi-l[X()
Xi+l]Xi+*
.” X, -= j(‘*)Xl
“’
Xi-l\Xf
) Xj+.l)X&*
“’ X1,, .
... .v-~K(x~) .X~+~ *.. X, when .X~ EL . (b) j(~~)xr ... xi .rx,axi+a ... xm = j(n)~l \I’e prove (a) ((b) is similar). Let yj == xj for j -/1 i, i -I- 1 and let yi r-1 .xi+, , yi-, = .vi . Then j(“).vr ... [xi , .+r] ... .r, = &s,,, j(.vr) -- (- l)“~‘*l-lf( ya,). Let the I) : s be written ID - K, where K -_= (C, ,..., C,,). Let Sk,, Cd,,., consist of those X? : s for which i and i -i- 1 belong to the same Cj and let wiw, x KLn - %n,71. WC may assume that i, i -L- 1 E C, when 3 E Si,,,n and that i E C.‘, , i 1. 1 E C, when a E Big,,, . Let 01’, a” bc the sum extended over %,,.n 9 fx,n 9 respectively. For a : = E E Bk,n we get C(E~ -= ECU and j(~c,) (-l)“~‘Llf(yc,) :-. j(..* xi , .xi+l . ..) - (-l)“Puf(... .yi+,xi . ..) : : j(...(.q, si+r) ...) and hence 0~’ 7: jar ... \xi , xi-,) ... x,,, . For 2 E %k,,l let a’ -... &‘, where (5:’ = (C; , Ci , C, ,..., C,) is defined by Ci 1:. C, with i I- 1 replaced bv i and CL -- C, with i replaced by i + 1. Then a” 7:. 3 and thus 23:,,,, is the disjoint union of sets of type {a, a’}. Let d = (Cg , C; , C:, ,..., C,). Then j(yc) f(y~,) --. f(yt) 1 q”j(y,.;) j(yc;) ... j(yc,,). \ve have E&’ : (-I)“~..P,-l~~‘,j(~C;) = /(xc,), j(-r,;) - j(.~,.~) and hence j(xz,) -- (-I)“~.“~ *!f(s9) so that (Y” 0. The proof of (a) is complete.
226
GUNNAR SJijDIN
We want to show that ~~1gives WL* the structure of a Hopf r-algebra if L E LIE’. First we show that it is a r-algebra. To do this it is obviously sufficient to find a r-algebra A for which there is an algebramonomorphism WL* --+gA, satisfying g(f lnl) = g(f)‘“). LEMMA
5. Let L E LIE’ be trivial. Rhen ~“1 gives WL* the structure of a
r-algebra. Proof. Give r(L*) the structure of a Hopf r-algebra by letting the elements of L* be primitive. Then Ir(L*) = L* @ Dr(L*) and hence pP(L*)* m L**. We get an isomorphism of graded vector spacesG: L -+ pr(L*)* defined by G(y)/L* = h(y), G(y)/DI’(L*) = 0. As is easily verified Pr(L*)* E LIE’ is trivial and hence G is an isomorphismin LIEf. Thus, using Theorem 1, we get an isomorphism a: r(L*) -“, r(L*)** -+(~O~G)’wL* of Hopf algebras.Now assumethat z E I’(L*)+ and let x1 ,..., x,,, EL. Put #?G(x)= @G(xJ ,..., ,8G(x,)). Then, by Lemma 3 and the fact that z is of even degree, we get a(ztn))(xl ..* x,) = @G(x,) -.* fiG(x,,,)) z(“) = J = ,&
WWM4
a(z) xa = a(z)[nl(xl a**:I).
W” Hence, a(z(“)) = a(z)l”l, which completesthe proof. Let WL* -+x EOWL be the algebra epimorphism in the proof of Theorem 1. Let {xi}iEN be a basisof L. Then x is given by xz, = %:I* E Ef,,WL. Let 2I be the set of finite sequencesr = (rI ,..., r,J in N such that ri < ri+i and ri < ri+l if de&x, ) is odd. Let x(r) = x,, ... x, E WL* and xr = x, .a*x, E WL. Then W%d ‘is a basisof WL@and xx(‘) z ?“‘* E pm,*WL. I; follok that {x+},,g generates WL as a graded vector spaceand we get an epimorphism of graded vector spacesF: WL+ --f WL given by F(x”‘) = x’. LEMMA 6. Let F be definedas above. Then F*: WL* --t (WL*)* is a monamorphismof algebrassatisfyingF*( j[nl) = F*( j)[“l and hence*Lnlgives WL.* the structure of a r-algebra.
Proof. Let r = (rl ,..., r,) E 6%and let yj = xii . For a subsequenceC of UP.., m) lety, be the product, formed asusual, in WL and pc the corresponding product in WL#. Then Fy, = ye . Thus ((F OF) 0 4 YL...~)
= (J’@FP)((Y,O~
+ 1 On).*.(~m@l
+ 1 OY~))
HOPF
which x’
=
ALGEBRAS
ASD
227
DEHIVATIOSS
shows that F is an epimorphism of coalgebras F* is a monomorphism * Consequently
since x(‘) -: y(l,...,m) and of algebras. Furthermore,
Y(1....,18!,
F”(f)P1
.V(l*....d -= DE F*(f)Lo
=y,&
h
tn..<
=f[“lYL..“d
m.n --
=
which
completes
rr~~~~~~
a functur (b) HOPF“‘. (c)
(f[“l
o 0
Yc ,.....
“d
= F”(f[“‘)
VCl,...,
n)
the proof.
2. (a) We hawe afumtor LIE’ --+ HOPFr~~kz HOPF’-’ - LIE’ +rn by A --+ PA*. These functors If L E LIE’
constitute
an equivalence
b-v L +
of the categories
WL* and LIE’
and
then HW,> = I?,* .
Proof. (a) It is obvious from the definition that I, -+ WL* is a functor from LIE! to the category of r-algebras and also to the category of Hopf algebras. To prove that it is a functor to HOPFrf we must show that the diagonal in WL* is a homomorphism of r-algebras. Consider the usual isomorphism of Hopf algebras n: lV(l, @L) --, WL @ WL given by V(X 01)) = .x @ 1 + 1 @ y for x,x G L. This induces an isomorphism of algebras wL*
@ WL” +
( WL (3 WI,) * 2-L
W(Z) @ I,) *.
Both WL* (3 WL* and W(L @L)* are r-algebras by the preceding results. \Ve want to show that x* o 5 is an isomorphism of r-algebras. The r-algebra structure on W,” @ WI,* is determined by the requirement that qi: WLx + WI* @ WLx, i --= 1, 2 are to be r-algebra homomorphisms, where ql(x) = .Y (3 1 and y2(x) = 1 @ x. Thus it is sufficient to show that rx o 5 0 qi , i = 1, 2 are homomorphisms of r-algebras. We have x* 2 5 o qi = W(pi)*, where p,: L c, 1, -+ I,, i == I, 2 are defined by pl(x @j y) 1 .r and p,(x @) y) : y. Since p, is a homomorphism in LW it follows that W(p,)* is a r-algebra homomorphism and hence so is ri* ” 5. Consider the diagram
where 9: is the multiplication map of WI. Sow, dW,.. -= an isomorphism of r-algebras. Thus it suffices to show algebra homomorphism. Let x1 ,..., xi F L @ 0 and Define p: I, (3 L --• L by P(.Y oy) = .Y + )‘. We put
5-l 0 F’ and xx 0 5 is that (p’o x)” is a T‘xi+, ,..., x,,, E 0 0 L. .V -= (xl ,..., x,,,) and
228
GUNNAR
SJijDIN
p(x) :. (p(x,),...,p(x,,,)). Note that elements of type x1 ... x, , with x as above, generate W(L @L) as a graded vector space and that (‘p 0 rr)xC = p(.+ if C is a subsequence of (I,..., m). Let f E WLT . Then
= f’“‘(p)
:: x)
x1 ‘.. x,
.-: ((v
0 7r)* ff”‘)
s,
... s,,, .
It follows that ((p, c r)*(f))t’ll = (~JIo rr)*flnl, i.e., (v o n)* is a r-algebra homomorphism and hence so is dwL.. This completes the proof that WL* E HOPF’-‘. (c) As in the proof of Theorem I we have an inequality H,, < Zi, . Consider the isomorphism A: WL + l WL”‘. Let x E L, g, h E ZWL*, s E WLT , and let n 3 2. Then (g . h)(.x) -- (g $> h)(x @) 1 + 1 @ x) ..- 0, d”](x)
=
e ;
‘&Xc,
... sxc,4
= 0 since in each term at least one .~e, = 1.
E 1.n Thus h(L) C fr(WZAx)* by Theorem I.
so that fiL < Z?P(~~.).
= fi,WL.,DWL.
= H,,.
=. H,,
(b) According to Theorem 1 and Lemma 3 A -c”” (M@A”)* is a natural isomorphism of Hopf r-algebras. By the reasoning in the proof of (c) the restriction of h to L gives a natural monomorphism I, + P( WL”)* of adjusted Lie algebras. However, Z?,- =. HW,- =: Z?ZJJ(~~.). and this implies that H,, = HptwL.). so that the mapping is an isomorphism. COROLLARY (Poincare-Birkhoff-Witt). Let L be an adjusted Lie algebra. Filter WL by the Lie filtration. Then the natural map xt: WL# -+ k.%WL is an isomorphism of algebras.
Proof. Obviously xL is an epimorphism of algebras. Assume first that L is connected of finite type. Then it is sufficient to show that HH.L e ::: HpwL , 2 we even have equality. Now let L be i.e., that fi, S: H,, , but by Theorem arbitrary. There is an exact sequence 0
-
l
Z,’ -
L(X)
+ L -+ 0
of adjusted Lie algebras, where L(X) is the free adjusted Lie algebra on a graded vector space X. Assume that we have proved the theorem for L(X). Then it follows (cf. .Milnor-Moore [3, p. 2511) that X,, is an isomorphism. Then, using the natural filtrations, we conclude that ZWL’ @ WL(X) --f WL is strict (i.e., im(ZWL’ (SJ WI, - Mx) n F,WL -: im(F,(ZW’L’ @ WL) --+ m)) and hence that E”WZ, -1 fl WZ,(X),!/Z?WZ,‘, which implies that x,. is an isomorphism.
HOPF
ALGEBRAS
.4ND
DERIVATIONS
229
Thus it is sufficient to show that xLtx) is an isomorphism. Using a direct limit argument we may assume that X is of finite type. We have W(X) = T(X), which is a bigraded algebra of finite type with T(X),,,
=: m<<-Yi,@
... 3 Xip,
i, + a-. + i,, = p 7. q,
and T(X),,, := k. It follows, as before, that Hopf r-algebra (allowing divided powers in is even and (p, 4) # (0,O)). Then, reasoning in two variables, we conclude that xttx) is an
WL(X)* :- T(X)= is a bigraded bidegrees (p, q) for which p + q as before, but with Hilbert series isomorphism.
KEFERENCES
I. M. ANDRE, Hopf algebras with divided powers, J Algebra 18 (1971), 19-50. 2. GIXIJKSEN-LEVIN, “Homology of Local Rings,” Queens Papers in Pure and Applied Mathematics, No. 20 (1969). 3. MII.NOR-MOORE, On the structure of Hopf algebras, Ann. o/Math. 81 (1965), 21 l-264.
OF ALGEBRA
64, 2 18-229
(1980)
Hopf
Algebras
and Derivations
Gl!NNAK
SJaDIN
Department of Mothemotics, University of Stockholm, Box 6701, S-113 85 Stockholm 23, Sweden Communicated
by D. Buchsbaum
Received May 9, 1978
Let A be a connected Hopf algebra of finite type over a field k. Then it is proved in Andre [I] that A* = UL, the universal enveloping algebra of a Lie algebra L, if and only if -4 is a Hopf algebra with divided powers. In [I] the concept of Lie coalgebras is introduced and the structure theorem above is proved and stated in its dual form. The main purpose of this paper is to prove this result directly. It is possible to give a short proof of the “if” part using a technique with derivations. This is done in Theorem 1. At the same time we prove that .4 is free as an algebra with divided powers, which extends the theorem of I,eray (cf. !Uilnor-I\‘Ioore [3, p. 2521) to arbitrary characteristics. In [I] it is supposed that there are divided powers in each dimension if char(k) : 2. Since the author is interested in applications to the Ext and Tor algebras of a local ring we do not use this definition for char(k) = 2. Instead we arc forced to introduce a slightly modified form of Lie algebras, which we call adjusted Lit algebras. If char(k) ;’ 2 th en an adjusted I,ie algebra is simply an ordinary I,ie algebra. We conclude the paper by getting the Poincare-Birkhoff-Witt theorem, for adjusted Lie algebras, as a corollary to the fact that the dual of the enveloping algebra is a Hopf algebra with divided powers.
%OTATIOSS
AND
CONVENTIONS
In the sequel k will denote an arbitrary field. All vector spaces, tensor products and direct sums are over K. We shall use the following symbols, definitions and conventions. I.
The sign # in a diagram
2.
In expressions
means that the diagram
for signs depending
218 0021-8693;80,050218-12$02.00/O Copyright 0 1980 by Academic Press, Inc. All rights of reproduction in any form reserved.
commutes.
on scariables given some degrees we
HOPF
ALGEBRAS
AND
DERIVATIONS
use the elements themselves to express the degrees, e.g., we write instead
of
3.
(-
219 (- I)a’b
])dWa).deW).
The letter N denotes the set of natural numbers or a finite part of it.
4. If X = @i>O Xi is a graded vector space of finite type then the Hilbert series H, is the formal power series defined by H,(z) = &,, (dim,X&?. If X,, = 0 we let fix denote the formal power series defined by $-1 Hx(z)
Obviously
=
g
“,I’-
Hx g,r = H, * H, and fixc,’
dlm,x,,-,
$ilm,x,.
= fix * fir .
5. For a graded vector space X we put X+ ~7 oi>l Xri and X- = Oi>Ox*i+~~
6. Suppose that X is a graded vector space. Then “x E X” always means that x is a homogeneous element of X, a basis of X means a homogeneous basis of X, etc. 7. If X, Y are graded vector spaces then X @ Y is a graded vector space with (X @ Y),, = @f+j,n Xi @ Yj and Hom,(X, Y) is a graded vector space with Hom,(X, Y)n = niPjcn Homk(Xi , Yj) = the set of k-linear maps f: X + Y such that f(Xi) C Yi-n . 8. As for signs we use a strict sign convention. This means that as soon as we define anything by means of a formula we should multiply by a sign related to the re-ordering of elements. The meaning of this principle is explained in the sequel. Let x = (x1 ,..., x,,J be a sequence of some elements, each of which is given a degree. Let (I E S(m) and put u(x) := (x0-r(r) ,..., x,-I(,,,)). Note that u(x) is obtained from x by moving the element in the ith place to the o(i)th place. It follows that (UT)(X) = u(T(x)) for (I, T E S(m). We put f(u, x) = (~)~~
220
GL’NNAR
SJijDIN
10. The commutator [ , ] in a graded algebra (meaning associative algebra) A is defined by [a, b] == a . b - (- l)a*bb . a. We say that A is commutative if [ , ] = 0 and that it is strictly commutative if furthermore a2 = 0 for a E A- . 11. A Y-algebra A is a strictly commutative graded algebra with divided powers of elements in A., , satisfying the condition of [I, p. 191. Note that we have the same definition if char(k) = 2, contrary to the case in [I]. We denote the exterior powers of x E A+ by x(n). For any .t E ,4 let x(O) == 1 and x(l) := x. If A is augmented, e.g., if A is connected, and IA is the augmentation ideal then we put DA Z*A $ &>s ky,(A+), where m(.r) = ,Y(“). If X is a graded vector space then Z’X denotes the free r-algebra on X. Note that H, = ZZr . 12. A Hopf r-algebra A is a Hopf algebra, which is also a r-algebra, such that d: A - ‘4 @ A is a homomorphism of Z-algebras. Using the universal property of the functor Z’ we see that if X is a graded vector space then there is a unique Hopf r-algebra structure on Z’X such that XC PI’X (actually there is then equality, which is an easy consequence, e.g., of Theorem 1). The category of connected Hopf r-algebras (of finite type) will be denoted by HOPFr (HOPF=‘). 13. Let X be a graded vector space. Then algebra on X (the tensor algebra). 14. An adjusted tural maps
Lie algebra I, is a graded
I, @L -‘L’L- -5 such that there is a monomorphism is a graded algebra, satisfying
7’(X)
is the free noncommutative
vector space L with two struc-
L of degree 0, L doubling
the degree
i: L -+ A of graded
vector spaces, where
A
and
L- --K-e
L
d # A- a~
Ii .4
If char(k) # 2 then an adjusted Lie algebra is just an ordinary Lie algebra. With the usual commutator and square any graded algebra may be considered as an adjusted Lit algebra. -4s in the usual case there is an enveloping algebra
HoPF
ALGEBRAS
AND
221
DERIVATIOSS
WL = TL/cII, where 2I is the ideal generated by elements [x, y] - i,x, y) for x, y EL and x2 - K(x) for x EL- . Note that L + WL is a monomorphism of adjusted Lie algebras. If X, Y are adjusted Lie algebras then so is X @ Y with <(x, y), (x’, y’)) = ((x, x’), (y, y’)) and K(x, y) = (KX, KY) and it follows by the universal property of W, that W(X @ Y) -+- WX @ WY, where (x, y) .. + x @ 1 - 1 6~ y for x E X, y E Y. It also follows that WL is a Hopf algebra with diagonal A: WL -Pd W(L @L) -+* WL @ WL, where d(x) r= (x, x) so that L C PWL. ,4n adjusted Lie algebra is said to be connected if L, = 0 and trivial if ,’ , ) and K are 0. The category of connected adjusted Lie algebras of finite type is denoted by LIE’. Sote that if L is trivial then WL is just the free strictly commutative algebra on the graded vector space L so that if L E LIE’ is trivial then H,, = Z^i, . If L is an adjusted Lie algebra then L* denotes the trivial adjusted Lie algebra with the same underlying graded vector space as I,.
DERIVATIONS
AND
THE
STRUCTURE
DEFISITIOK. Let A be a Z-algebra. derivation on A iff(xy) = f(x) . y -I- (The set of derivations on A is denoted In the following Hom,(A, A) is given
LEMMA
Hom,(A,
1.
OF ANDRE
Then f~ Hom,(A, A) is said to be a 1)““~ . f(y) andf(+)) = f(x) . xc” l). by Der A. an algebra structure via composition.
Let A he a r-algebra. Then Der A is a subadjusted Lie algebraof
A).
f, g
Proof.
Let
(a)
[f,g]
(b)
.fz E Der A if deg(
E Der A. We have to prove that
=fo.q--(-l)“~~cf~Der.4,
f)
(a) By direct computation [f, g](-y). Let x E A, . Then [f,gp'
THEOREMS
= [f,g](x)
is odd. we get v, g](xy)
*xc" -1) $ ((-l)"'"'"'g+~
and the fact that A is commutative Thus [f, g] E Der A.
= [f, g](x) . y + (- l)(~+O)‘s.~ *
.fx-(-])/'"7"'"4f~
.gx) . X(w-2)
and deg(x) is even makes the last term vanish.
(b) \Ve have f z(xy) = f'(x) * y - (( - 1)‘*(f+2) last term vanishes since deg( f) is odd.
-b (-
l)““)&x
* f y and the
Let x E A,. Thenf“x(n) zz f'(x) * ~('1-1) + (-l)"(fi")fx . fx * .~['+2) and fx .Jv = 0 since deg(fx) is odd and A is strictly commutative. It follows that f 2 E Der A and hence that Der A is an adjusted Lie algebra. The construction in the following lemma is basically the same one as in Gullikscn-Ixvin [2, p. 1091.
222
CUKNAR
LEMMA
SJGDIK
2. Let A E Hopfr and let for f E A* = Hom,(A, v(f) = j=E-4 -%
ABA
K)
‘@‘+A@k=A
Then, (a) A* -+” Hom,(A, A) is an algebra-ho momorphism (note that A* is always an algebra but not necessarily a Hopf algebra, unless A is of finite type); lb) (c)
= f; E Der A -f(DA)
v(f)/Ade,tf)
v(f)
=. 0;
(d) PA* = {f / f(DA) = 0} is a subadjusted Lie algebra of A* p?4* --ty Der A is a monomorphism of tijusted Lie algebras. Proof.
(a)
and
Let f, g E A* and consider the diagram A
LABA
1
A
A@A
#
&A
*@‘+A@A@A
We have 4g) o v(f)
= ((1 Og) o A) o ((1 of)
QA) = i +*
and
whence dg) 0 v( f ) = v(g * f ). (b)
This follows from the fact that d has a co-unit.
(c)
Suppose that f (DA) = 0. Then for x, y E IA
3(-v) = (1 c3f)4XY) = (1 Of)+)
=3(x>* y
= (1 Of)(&)
. (y 0 I) + (x 0 1>4YN
* (Y 0 1) + (-l)““(X + (-1px
0 I)((1 Of)4YN
.3(Y).
Suppose that x E A+ and let d(x) = x @ 1 + 1 @X + xs
c (x(Q) @ I)(1 @ x’““)(x; @x;)‘““’ **. (xi @ x:)‘“J. s,+...+s,=n.si*
HOPF
ALGEBRAS
AND
223
DERIVATIONS
But, (xi @ xp
=
if xi, xl are of odd degree and s > 2 O sy 0 .@) if x:, x; are of even degree
so it is sufficient to consider those terms for which si y= 0, 1 for 2 < follows that f,(n) :.= f(x) . Jn-1) .+ c
f($)
. J-1)
i <
Y.
It
. x;
3&q
==(lOf)(xOI+lOx+
c
x: @ A$ * (X-1)
@ 1) =i(x)
* Xc+1).
1
3(i
Now, suppose that v(f) E Der A. Then we have to show that f(DA) = 0. Let x, y E IA be such that xy E Adee . Then f(xy) = f(zy) = 3(x) * y + (--l)l’o~ .3(y) = 0 since deg(x), deg(y) < deg@). If x E A+ and rz 3 2 are such that x(n) E Adeg(,) then J+) = ffrcn) =3(‘(x) * @-I) == 0 by the same reason. Thusf(DA) = 0. (d) Let f,g E PA*. Then r$J g] = [df), v(g)] E Der A and hence u, g] E pAA* according to (c). Similarly, iff E PA* is of odd degree thenf* E PA*. Furthermore, v(f) = 0 implies that f = r( f )/Adeg(,) = 0, which shows that Y: PA* + Der A is a monomorphism of adjusted Lie algebras. I. (a) Let A E HOPFr. Then any section of IA -+ IA/DA induces ismorphismr(IA/DA) -+ a A of F-algebras. (b) If A E HOPFrf then PA* CA* inducesan ismorphismWfrA* -+c A* of Hopf algebrasand HA == I?,,,,, . THEOREM
an
Proof. Let A E HOPF r’. By the universal property of W we get a homomorphism/3 of algebrasasabove. Since PA* C PA* this is alsoa homomorphism of Hopf algebras.Let IA = DA @ X as a graded vector spaceand let {xJ~~,.,, where IV denotesthe set of positive integers (or a finite part of it), be a homogeneousbasis in X such that deg(x,) < deg(xj) implies that i < j. Consider the set ‘3 of sequencesY = (ri)isN of nonnegative integers such that ri = 0 for i > n(y) and such that ri = 0 or 1 if deg(xi) is odd. Order 2I by letting Y > Y’ if the last nonvanishing ri - r: > 0. Let fj E A* be given by -f;(p) =,fv and f,(DA @ k) = 0. For Y E 2I with rr = n(r) we put .@) = x,” a.0x1 ’ and f’=fi ***f 2. We have u(f’) = v(fJ’l o ..* o v(f,,>‘ll where the exponents now are with respectto composition. It follows that
v(f’) d”) = 0,
I
if if
Y>Y’
Y=
Y’
and in particular the sameis true for f rx(r’). Now, supposethat xreB &..z(r) = 0, where 23 is a finite nonempty subset of ‘3 and X, # 0 for I E!B. Let s be the
223
CUNNAR
SJijDIN
largest element in 23. Then 0 -= f Yxres /\,.T(~) = h, , which is a contradiction. Hence the x(‘): s arc linearly independent and since, by induction on the degree, it is seen that they generate A it follows that they constitute a basis for A. In particular XC A induces an isomorphism of r-algebras rX - A, which proves (a) when A is of finite type. In the general case TX --f A is an epimorphism by the same argument as above. To prove that it is a monomorphism it is enough to notice that any finite-dimensional vector space I/C X is contained in a sub-Hopf r-algebra of finite type. Another possibility would be to repeat the former argument but now using a well-ordered basis. Kow, let us turn to the proof of (b). According to the formula above {f r},Ell is expressed in the dual basis of {x(‘)} rest by a triangular matrix with 1 : s in the diagonal and hence (f r},es is a basis for A. In particular {fi}ieN generates A* as an algebra. Since fi E PA* this implies that fl is an epimorphism. By (a) HA. = HA : HrcIAiDAj = It?,,, ,DA = - Z?JS~,(this shows the formula in (b)) and hence it will suffice to show that HH.pA. ,< Z?p,. . We prove that H,, ,< flL for anyL E LIE’. Filter WL by F, WL = the elements of WL that may be written as polynomials of degree
Let PI,,,,, be the set of sequences (E -= (C, ,..., C,,), where
(a) Ci is a subsequence, possibly empty, of (I,..., m). (b) The underlying subsets of the Ci : s are disjoint and their union is {I,..., m}. Let f%,, be the set of sets D =- {C, ,..., C,}, where the Ci : s satisfies (a) and (b) above. Note that we have a surjective mapping PI,., --f 9,,*, given by 0. = (C, ,...) C,) + E = {C, ,...) C,}. As usual, if T E S(n), we put ~((5) = Then oe E S(m) is given by (C’(1) ,*.a, C,-,(,j). Let Cf = (rp+r).,r ,..., yvti)). u$(j) I= rj . Let A be a commutative algebra and let x == (x1 ,..., x,,,), where xi E A. Then, if C 7. (il ,..., is) is a subsequence of (I,..., m), we put x, = xi, ... xi8 (=:I if C is empty). In particular xc1,..., m) = xr ..’ x,, . For 0. E ‘& .n we put cc2 = E(U~ , x). Ixtf, )...(f,?nE Hom,(A, A) and let y E A+ . Then, for (5 -= (C, ,..., C,) E ‘?I,,,, , let f&) : - cE’fc,(y) ... . fc,(y). Using the fact that y has even degree and A is commutative we get f,(o)(y) = fa(-v) for T E s(n). Thus we may define f&v), for a c %,,., , by the formula f%(y) -=f&(T), whcrc 0. E 91,,,, is such that c - II).
HOPF
ALGEBRAS
Let A E HOPFr
AND
225
DERIVATIOXS
and let x1 ,..., x,, E PA*, y E PA”,
LEMMA
3.
I’rooj.
It suffices to show that
y E A+ . Then
where x =_ (X r ,..., .v,,,), and since the Si : s are derivations this follows by induction on m. Now, dually, let A be an algebra of finite type and letj E AT and x1 ,..., x,,, E A. Let, for 0: E PI,n., , j(xe) = g~j(xc-) . ..**j(~c.). Then, since im(j) CK, k is commutative and j has even degree, we get j(+tc,) = j(xc). It follows that we may define j(+) for 3 =- c E d,,, by putting j(xn) :.: j(~). Inspired by Lemma 3 we prove
l
LEMMA wL,T.,,,,J,
4. Let I, E LIEJ and let f E WLZ . Then there is a unique j[“l which satisfies f [nJx,... x,,,= xFEs,., j(xx) for x1 ,..., .Y,~EL.
E
Proof. Obviously jl”l is unique and jtnl E WZ4z.dey(J, if it exists. Define jcrL) E (TL)” by j(n)~i ... x,,, = &.a,., j(x,), where Xi EL, xl .*. X, is formed in TL and the x a : s are formed with respect to the product in WL. It will then be sufficient to show that the ideal, generated by elements of type [x, y] - (x, y) with X, y EL or x2 - K(X) with .v EL , is mapped into 0 by j(“). Thus we have to show that (a)
j(R).~,
“’
Xi-l[X()
Xi+l]Xi+*
.” X, -= j(‘*)Xl
“’
Xi-l\Xf
) Xj+.l)X&*
“’ X1,, .
... .v-~K(x~) .X~+~ *.. X, when .X~ EL . (b) j(~~)xr ... xi .rx,axi+a ... xm = j(n)~l \I’e prove (a) ((b) is similar). Let yj == xj for j -/1 i, i -I- 1 and let yi r-1 .xi+, , yi-, = .vi . Then j(“).vr ... [xi , .+r] ... .r, = &s,,, j(.vr) -- (- l)“~‘*l-lf( ya,). Let the I) : s be written ID - K, where K -_= (C, ,..., C,,). Let Sk,, Cd,,., consist of those X? : s for which i and i -i- 1 belong to the same Cj and let wiw, x KLn - %n,71. WC may assume that i, i -L- 1 E C, when 3 E Si,,,n and that i E C.‘, , i 1. 1 E C, when a E Big,,, . Let 01’, a” bc the sum extended over %,,.n 9 fx,n 9 respectively. For a : = E E Bk,n we get C(E~ -= ECU and j(~c,) (-l)“~‘Llf(yc,) :-. j(..* xi , .xi+l . ..) - (-l)“Puf(... .yi+,xi . ..) : : j(...(.q, si+r) ...) and hence 0~’ 7: jar ... \xi , xi-,) ... x,,, . For 2 E %k,,l let a’ -... &‘, where (5:’ = (C; , Ci , C, ,..., C,) is defined by Ci 1:. C, with i I- 1 replaced bv i and CL -- C, with i replaced by i + 1. Then a” 7:. 3 and thus 23:,,,, is the disjoint union of sets of type {a, a’}. Let d = (Cg , C; , C:, ,..., C,). Then j(yc) f(y~,) --. f(yt) 1 q”j(y,.;) j(yc;) ... j(yc,,). \ve have E&’ : (-I)“~..P,-l~~‘,j(~C;) = /(xc,), j(-r,;) - j(.~,.~) and hence j(xz,) -- (-I)“~.“~ *!f(s9) so that (Y” 0. The proof of (a) is complete.
226
GUNNAR SJijDIN
We want to show that ~~1gives WL* the structure of a Hopf r-algebra if L E LIE’. First we show that it is a r-algebra. To do this it is obviously sufficient to find a r-algebra A for which there is an algebramonomorphism WL* --+gA, satisfying g(f lnl) = g(f)‘“). LEMMA
5. Let L E LIE’ be trivial. Rhen ~“1 gives WL* the structure of a
r-algebra. Proof. Give r(L*) the structure of a Hopf r-algebra by letting the elements of L* be primitive. Then Ir(L*) = L* @ Dr(L*) and hence pP(L*)* m L**. We get an isomorphism of graded vector spacesG: L -+ pr(L*)* defined by G(y)/L* = h(y), G(y)/DI’(L*) = 0. As is easily verified Pr(L*)* E LIE’ is trivial and hence G is an isomorphismin LIEf. Thus, using Theorem 1, we get an isomorphism a: r(L*) -“, r(L*)** -+(~O~G)’wL* of Hopf algebras.Now assumethat z E I’(L*)+ and let x1 ,..., x,,, EL. Put #?G(x)= @G(xJ ,..., ,8G(x,)). Then, by Lemma 3 and the fact that z is of even degree, we get a(ztn))(xl ..* x,) = @G(x,) -.* fiG(x,,,)) z(“) = J = ,&
WWM4
a(z) xa = a(z)[nl(xl a**:I).
W” Hence, a(z(“)) = a(z)l”l, which completesthe proof. Let WL* -+x EOWL be the algebra epimorphism in the proof of Theorem 1. Let {xi}iEN be a basisof L. Then x is given by xz, = %:I* E Ef,,WL. Let 2I be the set of finite sequencesr = (rI ,..., r,J in N such that ri < ri+i and ri < ri+l if de&x, ) is odd. Let x(r) = x,, ... x, E WL* and xr = x, .a*x, E WL. Then W%d ‘is a basisof WL@and xx(‘) z ?“‘* E pm,*WL. I; follok that {x+},,g generates WL as a graded vector spaceand we get an epimorphism of graded vector spacesF: WL+ --f WL given by F(x”‘) = x’. LEMMA 6. Let F be definedas above. Then F*: WL* --t (WL*)* is a monamorphismof algebrassatisfyingF*( j[nl) = F*( j)[“l and hence*Lnlgives WL.* the structure of a r-algebra.
Proof. Let r = (rl ,..., r,) E 6%and let yj = xii . For a subsequenceC of UP.., m) lety, be the product, formed asusual, in WL and pc the corresponding product in WL#. Then Fy, = ye . Thus ((F OF) 0 4 YL...~)
= (J’@FP)((Y,O~
+ 1 On).*.(~m@l
+ 1 OY~))
HOPF
which x’
=
ALGEBRAS
ASD
227
DEHIVATIOSS
shows that F is an epimorphism of coalgebras F* is a monomorphism * Consequently
since x(‘) -: y(l,...,m) and of algebras. Furthermore,
Y(1....,18!,
F”(f)P1
.V(l*....d -= DE F*(f)Lo
=y,&
h
tn..<
=f[“lYL..“d
m.n --
=
which
completes
rr~~~~~~
a functur (b) HOPF“‘. (c)
(f[“l
o 0
Yc ,.....
“d
= F”(f[“‘)
VCl,...,
n)
the proof.
2. (a) We hawe afumtor LIE’ --+ HOPFr~~kz HOPF’-’ - LIE’ +rn by A --+ PA*. These functors If L E LIE’
constitute
an equivalence
b-v L +
of the categories
WL* and LIE’
and
then HW,> = I?,* .
Proof. (a) It is obvious from the definition that I, -+ WL* is a functor from LIE! to the category of r-algebras and also to the category of Hopf algebras. To prove that it is a functor to HOPFrf we must show that the diagonal in WL* is a homomorphism of r-algebras. Consider the usual isomorphism of Hopf algebras n: lV(l, @L) --, WL @ WL given by V(X 01)) = .x @ 1 + 1 @ y for x,x G L. This induces an isomorphism of algebras wL*
@ WL” +
( WL (3 WI,) * 2-L
W(Z) @ I,) *.
Both WL* (3 WL* and W(L @L)* are r-algebras by the preceding results. \Ve want to show that x* o 5 is an isomorphism of r-algebras. The r-algebra structure on W,” @ WI,* is determined by the requirement that qi: WLx + WI* @ WLx, i --= 1, 2 are to be r-algebra homomorphisms, where ql(x) = .Y (3 1 and y2(x) = 1 @ x. Thus it is sufficient to show that rx o 5 0 qi , i = 1, 2 are homomorphisms of r-algebras. We have x* 2 5 o qi = W(pi)*, where p,: L c, 1, -+ I,, i == I, 2 are defined by pl(x @j y) 1 .r and p,(x @) y) : y. Since p, is a homomorphism in LW it follows that W(p,)* is a r-algebra homomorphism and hence so is ri* ” 5. Consider the diagram
where 9: is the multiplication map of WI. Sow, dW,.. -= an isomorphism of r-algebras. Thus it suffices to show algebra homomorphism. Let x1 ,..., xi F L @ 0 and Define p: I, (3 L --• L by P(.Y oy) = .Y + )‘. We put
5-l 0 F’ and xx 0 5 is that (p’o x)” is a T‘xi+, ,..., x,,, E 0 0 L. .V -= (xl ,..., x,,,) and
228
GUNNAR
SJijDIN
p(x) :. (p(x,),...,p(x,,,)). Note that elements of type x1 ... x, , with x as above, generate W(L @L) as a graded vector space and that (‘p 0 rr)xC = p(.+ if C is a subsequence of (I,..., m). Let f E WLT . Then
= f’“‘(p)
:: x)
x1 ‘.. x,
.-: ((v
0 7r)* ff”‘)
s,
... s,,, .
It follows that ((p, c r)*(f))t’ll = (~JIo rr)*flnl, i.e., (v o n)* is a r-algebra homomorphism and hence so is dwL.. This completes the proof that WL* E HOPF’-‘. (c) As in the proof of Theorem I we have an inequality H,, < Zi, . Consider the isomorphism A: WL + l WL”‘. Let x E L, g, h E ZWL*, s E WLT , and let n 3 2. Then (g . h)(.x) -- (g $> h)(x @) 1 + 1 @ x) ..- 0, d”](x)
=
e ;
‘&Xc,
... sxc,4
= 0 since in each term at least one .~e, = 1.
E 1.n Thus h(L) C fr(WZAx)* by Theorem I.
so that fiL < Z?P(~~.).
= fi,WL.,DWL.
= H,,.
=. H,,
(b) According to Theorem 1 and Lemma 3 A -c”” (M@A”)* is a natural isomorphism of Hopf r-algebras. By the reasoning in the proof of (c) the restriction of h to L gives a natural monomorphism I, + P( WL”)* of adjusted Lie algebras. However, Z?,- =. HW,- =: Z?ZJJ(~~.). and this implies that H,, = HptwL.). so that the mapping is an isomorphism. COROLLARY (Poincare-Birkhoff-Witt). Let L be an adjusted Lie algebra. Filter WL by the Lie filtration. Then the natural map xt: WL# -+ k.%WL is an isomorphism of algebras.
Proof. Obviously xL is an epimorphism of algebras. Assume first that L is connected of finite type. Then it is sufficient to show that HH.L e ::: HpwL , 2 we even have equality. Now let L be i.e., that fi, S: H,, , but by Theorem arbitrary. There is an exact sequence 0
-
l
Z,’ -
L(X)
+ L -+ 0
of adjusted Lie algebras, where L(X) is the free adjusted Lie algebra on a graded vector space X. Assume that we have proved the theorem for L(X). Then it follows (cf. .Milnor-Moore [3, p. 2511) that X,, is an isomorphism. Then, using the natural filtrations, we conclude that ZWL’ @ WL(X) --f WL is strict (i.e., im(ZWL’ (SJ WI, - Mx) n F,WL -: im(F,(ZW’L’ @ WL) --+ m)) and hence that E”WZ, -1 fl WZ,(X),!/Z?WZ,‘, which implies that x,. is an isomorphism.
HOPF
ALGEBRAS
.4ND
DERIVATIONS
229
Thus it is sufficient to show that xLtx) is an isomorphism. Using a direct limit argument we may assume that X is of finite type. We have W(X) = T(X), which is a bigraded algebra of finite type with T(X),,,
=: m<<-Yi,@
... 3 Xip,
i, + a-. + i,, = p 7. q,
and T(X),,, := k. It follows, as before, that Hopf r-algebra (allowing divided powers in is even and (p, 4) # (0,O)). Then, reasoning in two variables, we conclude that xttx) is an
WL(X)* :- T(X)= is a bigraded bidegrees (p, q) for which p + q as before, but with Hilbert series isomorphism.
KEFERENCES
I. M. ANDRE, Hopf algebras with divided powers, J Algebra 18 (1971), 19-50. 2. GIXIJKSEN-LEVIN, “Homology of Local Rings,” Queens Papers in Pure and Applied Mathematics, No. 20 (1969). 3. MII.NOR-MOORE, On the structure of Hopf algebras, Ann. o/Math. 81 (1965), 21 l-264.