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INVARIANT THEORY AND THE INVARIANTS OF LOW-DIMENSIONAL TOPOLOGY
Topology Vol. 38, No. 4, pp. 763—777, 1999 1999 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0040-9383/99 $19.00#0.00
PII: S0040-9383(98)00014-7
INVARIANT THEORY AND THE INVARIANTS OF LOW-DIMENSIONAL TOPOLOGY KEQIN LIU (Received 27 December 1996; in revised form 2 December 1997; accepted 20 January 1998)
1. INTRODUCTION
The aim of this paper is to disclose the connection between invariant theory and the invariants of low-dimensional topology. The bridge which connects the two subjects is based upon the amazing connection between the deep algebraic structures of Hopf algebras and Kauffman—Radford—Hennings invariants of 3-manifolds, where Kauffman—Radford— Hennings invariants of 3-manifolds are a new class of 3-manifold invariants constructed by employing the right integrals on certain Hopf algebras in [4—6]. Since Kauffman—Radford— Hennings invariants of 3-manifolds are just some scalar multiples of Kauffman’s knot invariants and every trace on a quantum algebra produces a Kauffman’s knot invariant, so the study of traces on quantum algebras plays a key role in the study of Kauffman—Radford—Hennings invariants of 3-manifolds. In this paper, we will prove that determining the traces on a quantum Hopf algebra is equivalent to determining a ‘‘homogeneous’’ part of the algebra of invariants of a group acting on an associative algebra ZM . This fact not only joins the classical invariant theory to Kaufman’s knot invariants in the case that ZM is commutative, but also shows us the necessity of considering the invariants theory for non-commutative algebras in the case that ZM is not commutative. The invariant theory for some non-commutative algebras like exterior algebras and superalgebras has been studied in [2, 14] for different motivations. The connection between invariant theory and the invariants of low-dimensional topology provides another motivation of studying invariant theory for non-commutative algebras. Hopefully, invariant theory can eventually arise again from the realm of non-commutative algebras. This paper is organized as follows. In Section 2 we sketch the construction of Kauffman’s knot invariants and exhibit the connection between Kauffman’s knot invariants and quantum algebras with a trace. In Section 3 we recall some facts in classical invariant theory and some properties about integrals on finite-dimensional Hopf algebras. In Section 4 we generalize semi-products in group theory and introduce the notion of the map-product of two groups. The notion of map-products, which is of independent interest, is the essential part in establishing the connection between invariant theory and the study of the traces on quantum Hopf algebras. In Section 5 we construct the nice algebra ZM " : ZM (u) from a quantum Hopf algebra H and a group-like element u3H with the property that the elements of a group G act on ZM as either algebra automorphisms or anti-automorphisms. Therefore, the set ZM % " : +z3 ZM "x(z)"z for all x3G, forms of Jordan subalgebra of ZM >, where ZM > is the Jordan algebra obtained from the associative algebra ZM . ZM % is called the Jordan algebra of invariants of the group G. In 763
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Section 6 we prove that determining the traces on the quantum algebra H is the same as determining a ‘‘homogeneous’’ part of the Jordan algebra ZM % of invariants of the group G.
2. KAUFFMAN’S KNOT INVARIANTS
In this section we explain the topological background of studying quantum algebras. Since it is not necessary to use invariants that depend on link orientation to define 3-manifold invariants, we consider only the invariants for unoriented links. Kauffman proves that an invariant of regular isotopy of unoriented knots can be constructed from a quantum algebra with a trace, where quantum algebras are a class of associative algebras introduced by Kauffman in [4]. Definition 2.1. An associative algebra H over a commutative ring k is called a quantum algebra if there exist S3Hom (H, H), R3 H H and G 3 H such that I (1) S(xy)"S(yx) for x, y3 H, (2) R R R "R R R , (3) R\"(S 1)(R)"(1 S\)(R), (4) S(G)"G\ and S(x)"GxG\ for x3 H, where R " : R 13 H H H, R :"(1 p)(R ), R " : (p 1)(R ) and p is the twist automorphism of H H given by p : x y > y x for x, y 3 H. The quantum algebra H depending on S, R and G is denoted by (H, S, R, G). A useful corollary of (3) in Definition 2.1 is the equations (SK SK)(R$)"R$ for m3 9
(2.1)
Definition 2.2. Let (H, S, R, G) be a quantum algebra. A functional tr 3 H* " : Hom (H, k) is said to be a trace on H if it satisfies the following properties: I (1) tr(xy)"tr(yx) for x, y3 H, (2) tr(x)"tr(S(x)) for x 3 H. There are four basic building blocks in a knot diagram with respect to the vertical direction (of the paper). They are minima, maxima and two types of crossing:
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For example, the trefoil knot has two maxima, two minima and three crossings:
The four kinds of basic building blocks of the trefoil knot can be seen clearly if we use some vertical line segments in its diagram as follows:
This kind of description of a knot diagram is convenient in the construction of Kauffman’s knot invariants. Let (H, S, R, G) be a quantum algebra with a trace tr3 H*, where R" e f and G G G e , f 3 H. We call e f the general term of R with the sum index i. By Definition 2.1 (3), G G G G both S(e ) f and e S\( f ) is the general term of R\. Given a knot diagram K, we G G G G decorate the crossings of K by using the general term of R or R\ in the following ways:
In order to indicate different crossings, we use different sum index for the general term of R or R\. For example, the diagram of trefoil knot can be decorated as
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where we have used three different indices i, j and k to indicate the three crossings of the trefoil knot. We think of the knot diagram K as a closed loop of wire around which beads can be moved, and regard elements of the quantum algebra H appearing in the knot diagram K as beads. Choose a point on a vertical line segment in the diagram which is not part of a crossing. Moving upward from that point, slide all beads around the diagram into the vertical segment. As a bead moves across maxima or minima its label will change according to the following rule:
The final juxtaposition of labelled beads at the chosen vertical line segment gives rise to a product in H with finitely many sum indices (down to up is interpreted as multiplication left to right). Using the product as the general term, we get a sum w in H, where the sum is taken over all different sum indices appearing in the product. Let d be the Whitney degree of the diagram K that is obtained by traversing K upward from the chosen vertical line segment, where the beads have been concentrated. Then Kauffman’s knot invariant for K is defined by ¹R(K) :"tr(GBw). For example, if we move all beads in Fig. 1 from the point indicated by ;, then
Fig. 1.
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and d"!2. The sum w in H with the general term: S(e ) S( f )S(e )S( f )S(e ) f is G H I G H I w" S(e )S( f )S(e )S( f )S(e ) f . G H I G H I G H I The Kauffman’s knot invariant for the trefoil knot is ¹R(K)" tr (G\S(e )S( f )S(e )S( f )S(e ) f ) G H I G H I G H I which can be rewritten as ¹R(K)" tr(G\S( f )S(e )S( f )e f e ) G H I G H I G H I by (2.1). Note that the value of ¹R(K) is independent of the choice of vertical segment along the diagram for K at which the beads have been concentrated.
3. PRELIMINARIES ABOUT INVARIANT THEORY AND HOPF ALGEBRAS
For convenience, from this section we will assume that algebras and coalgebras are over a field k with char(k)O2. First, let us recall some basic facts in classical invariant theory. Let R"k[a , 2 , a ] be L a finitely generated commutative algebra over k, where the a ’s may be not algebraically G independent over k. Suppose that a group G acts on R by algebra automorphisms. An element f of R is called an invariant of G if x( f )"f for all x 3 G. The set consisting of all invariants of G is called the ring of invariants and denoted by R%, i.e. R%"+ f 3R"x( f )"f for all x3G,. Clearly, R% is a subalgebra of R. The basic object of study of invariant theory is the ring of invariants R%. Many interesting theorems have been found in order to describe R%. For our purpose, we need a theorem proved by Richman in [13]. PROPOSITION 3.1. Suppose that a finite solvable group G acts on a finitely generated commutative k-algebra R"k[a , 2 , a ]. If the order "G" of G is not divisible by the L characteristic of the field k, then the ring of invariants R% is generated over k by x(aG aG 2 aGL), L V Z% where i , 2 , i *0 and i #2#i )"G". L L Proof. Proposition 5 in [13].
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Now let us recall some properties about integrals on finite-dimensional Hopf algebras. Let (H, *, e, S) be a finite-dimensional Hopf algebra with comultiplication *, counit e and antipode S. Let H* " : Hom (H, k) be the linear dual of H. Then H* is also a Hopf algebra. I We use G(H) to denote the set consisting of all groupslike elements of H. Then G(H*)"Alg (H, k); that is, G(H*) consists of all algebra homomorphisms from the algebra I H to the algebra k.
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Let " be a non-zero left integral for H. Since the ideal of left integrals for H is one-dimensional. Hence, there exists a unique a3 G(H*) such that "h"a(h)" for h3 H.
(3.1)
Similarly, let j be a non-zero right integral for H*, then exists a unique g3 G(H) such that pj"p(g)j for p3 H*.
(3.2)
Definition 3.1. Let " be a non-zero left integral for H and j be a non-zero right integral for H*. Then the unique group-like element a satisfying (3.1) is called the distinguished group-like element of H*, and the unique group-like element g satisfying (3.2) is called the distinguished group-like element of H. H* is a H-bimodule under the following actions: (h-p)(x) " : p(xh),
(p ¤h)(x) " : p(hx),
and H is a H*-bimodule under the following actions: p-h " : h p(h ), h¤p " : p(h )h , F F where h, x3H and p3 H*. PROPOSITION 3.2. ¸et (H, *, e, S) be a finite-dimensional Hopf algebra and j a non-zero right integral for H*. ¸et g be the distinguished grouplike element of H and let a be the distinguished grouplike element of H*. (1) (H*, ¤) is a free right H-module with basis j. (H*, -) is a free left H-module with basis j. (2) j(xy)"j(S(y¤a)x) for x, y3H. (3) j(S(x))"j(gx) for x 3 H. (4) Radford’s formula holds: S(x)"g(a-x¤a\)g\ for x 3H. (5) H is unimodular if and only if a"e. (6) H* is unimodular if and only if g"1. Proof. The proof of (1)—(4) can be found in [11]. (5) and (6) are clear.
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4. MAP-PRODUCTS OF TWO GROUPS
Let K and ¼ be two groups. We use map(¼, ¼) to denote the set consisting of all maps from ¼ to ¼. Definition 4.1. A map u : K;KPmap(¼, ¼) is called a group-producing map if u satisfies the following conditions: (1) For k , k , k 3K and w , w 3 ¼, u(k k , k )[u(k , k )(w )w ]"u(k , k k )(w )u(k , k )(w ).
(4.1)
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(2) There exists an element w of ¼ such that u(1 , k)(w )"1 , u(k, 1 )(w)"ww\ for k3 K and w 3¼. ) 5 )
(4.2)
PROPOSITION 4.1. Given two groups K and ¼ and a map u : K;KPmap(¼, ¼). ¹hen the following are equivalent: (1) u is a group-producing map. (2) ¼;K is a group under the product (w , k )(w , k ) " : (u(k , k )(w )w , k k ) for w 3¼ and k 3 K, G G
(4.3)
where (w , 1 ) is the identity and (w, k)\"(u(k, k\)(w)\w , k\) for w3 ¼ and k 3 K. ) )
Proof. A direct calculation.
Definition 4.2. The group defined by (4.3) is called the map-product of ¼ by K via the map u. Example. Let ¼ and K be two groups. Let t : KPAut(¼) be a group homomorphism. Define a map u : K;KPmap(¼, ¼) by u(k , k )(w) " : wRI for w 3 ¼ and k , k 3K. Then it is clear that u is a group-producing map, and the map-product of ¼ by K via the map u is the semi-direct product of ¼ by K. Remark Like the semi-direct product, the map-product of ¼ by K contains a normal subgroup which is isomorphic to the group ¼. Another important special case of the map-product of ¼ by K is from some central elements of the group ¼. Let x be an element of a group. We set
"x" " :
the order of x if the order of x is finite 0 if the order of x is infinite.
PROPOSITION 4.2. Given two groups ¼ and K and a map ¹ : K;KP9/"h"9, where h is in the center of the group ¼. ¹hen ¼;K becomes a group under the product: (w , k )(w , k ) " : (w w h2I I, k k )
(4.4)
if and only if the map ¹ satisfies the conditions: ¹(k , k )#¹(k k , k ),¹(k , k k )
(mod"h"),
¹(1 , k),¹(k, 1 ) (mod"h"), ) ) where w , w 3¼ and k, k , k , k 3 K. Proof. Define a map u : K;KPmap(¼, ¼) by u(k , k )(w) " : wh2I I
for k , k 3 K and w3 ¼.
(4.5) (4.6)
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Then it is easy to check that u is a group-producing map if and only if ¹ satisfies (4.5) and (4.6). So Proposition 4.2 follows from Proposition 4.1. ) Definition 4.3. The map-product of ¼ by K defined in Proposition 4.2 is called the central map-product of ¼ by K determined by h and ¹. We finish this section by giving a sufficient condition for the claim that a group G can act on the central map-product of ¼ by K determined by h and ¹. PROPOSITION 4.3. Given three groups G, ¼ and K and a map ¹ : K;KP9/"h"9, where h is in the center of the group ¼ and ¹ satisfies (4.5) and (4.6). ¸et u : GPAut(¼) be a group homomorphism such that u(x)(h)"h V for some group homomorphism * : GP4(9/"h"9), where 4,(9/"h"9) is the unit group of (9/"h"9). (1) If a map t : GP9/"h"9 has the property t(xy),t(x)#*(x)t(y) (mod"h")
for x, y, 3G,
then the group G can act on the central map-product of ¼ by K determined by h and ¹ in the following way: x ° (w, k)"(u(x)(w)h2I IR V, k) for x3 G, w 3 ¼ and k3 K.
(4.7)
(2) If t satisfies both (4.7) and ¹(k , k )(*(x)!1)"(¹(k , k )#¹(k , k )!¹(k k , k k ))t(x), where k , k 3K and x 3G, then the action defined by (4.7) is an automorphism of the central map-product of ¼ by K determined by h and ¹.
5. THE NICE SUBALGEBRAS OF HOPF ALGEBRAS
In this section we assume that (H, S, R, G) is a finite-dimensional Hopf algebra over a field k, where char(k)"0. We will construct the nice subalgebras from H and grouplike elements of H, and prove that a cyclic subgroup of G¸(H) acts on the nice subalgebras naturally as either algebra automorphisms or anti-automorphisms. LEMMA 5.1. ¸et g3G(H*) and u3G(H) (1) Both g- and g¤ is an algebra automorphism of H. (2) For any h 3 H, we have g-(uhu\)"u(g-h)u\, (uhu\)¤g"u(h¤g)u\ S(g-h)"g-S(h), S(h¤g)"S(h)¤g.
(5.1) (5.2)
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Proof. (1) For x, y 3 H we have (xy)¤g" g(x y )x y V W
" g(x )x g(y )y "(x¤g)(y¤g). V W Similarly, one can check that g- is an algebra automorphism of H. (2) (5.1) follows from (1) and (5.2) is clear.
)
Let u 3 G(H) and a be the distinguished group-like element of H*. Define A, B, º 3 Aut(H) by A(h) " : h¤a, B(h) " : a\-h, º(h) " : uhu\ where h 3H. We use ¼ to denote the subgroup of Aut(H) generated by A, B and º. It is clear that ¼ is an abelian group and ¼" : +AKBLºI"m, n, k 3 9,. LEMMA 5.2. For any m, n, k 3 9, we have (AKBL)(h)"aK\L(h)h for h3G(H). S\AKBLºIS"ALBKºI.
(5.3) (5.4)
Proof. (5.3) is clear. (5.4) follows from the following two identities: S\AKS"BK for m 3 9
(5.5)
S\BLS"AL for n39.
(5.6) )
For w3¼ and a3 9 " : +0, 1,, we set " : +u?z"zx"w(x)z for x3 H,, Z U ? Z" Z for a 3 9 , ? U ? UZ5 Z(u) " : Z #Z . It is easy to check that 1!(!1)??Y ¹(a, a ) " : for a, a 3 9 2 is a map form 9 ;9 to 9/"º"9 satisfying both (4.5) and (4.6). By Proposition 4.2, we have the central map-product of ¼ by 9 determined by º and ¹ whose group operation * is given by (w, a) * (w , a )"(ww º2? ?Y, a#a ) for w, w 3 ¼ and a, a 39 . PROPOSITION 5.3. ¹he associative algebra Z(u) has the following properties: (1) Z(u) is ‘‘graded’’ by the central map-product of ¼ by 9 determined by º and ¹ in the sense: Z Z -Z "Z , (5.7) U ? UY ?Y U ? * UY ?Y UUY32? ?Y ?>?Y where w, w 3 ¼ and a, a 3 9 .
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(2) ¸et r"dim Z . ¹here exists a linear basis +z , 2 , z , of Z (-8 Z ) and P U Z 5 U r irreducible character s , 2 , s of the abelian group ¼ such that Z(u) is spanned P linearly by +u?z "a3+0, 1,, 1)i)r, and G (u?G z )(u?H z )"aKG\LG (u?H)aLH\KH (u?G)s (w )(u?H z )(u?G z ), G H H G H G where 1)i, j)r, a 39 , z 3Z - and w "AKGBLGºIG 3¼ for some m , n , k 39. G G UG G G G G (3) For v, t, m, n, k39 and a3 9 , (AS\\T)R(Z K L I )"Z \R> K>\R\ L 3 ?
\R> L>\R\ K \RI>\R\ \\?
3
. ? (5.9)
Proof. (1) For convenience, we use z to indicate that z is an element of Z . So U U U z x"w(x)z for all x3 H and z z 3 Z . By (5.3), we get U U U UY UUY (u?z )(u?Yz )"aK\L(u?Y)u?>?Yz z , U UY U UY
(5.10)
where a, a 3 9 , w, w 3 ¼ and w"AKBLºI for some m, n, k39. It follows from (5.10) that (5.7) is true if one of a and a is 0. In order to prove that (5.7) is also true for a"a "1, it suffices to prove that uz z 3 Z , which holds by the following: U UY UUY3 (uz z )x"u(z z x)"uww (x)z z "uww (x)u\uz z U UY U UY U UY U UY "(ºww )(x)(uz z ) for x 3H. U UY (2) Since Z is ¼-invariant and ¼ is finite, Z is a complete reducible ¼-module U U for every w 3¼. This implies that there exists a ¼-submodule Z of Z such that U U Z " Z . Expressing every non-zero Z as a direct sum of one-dimensional U Z 5 U U irreducible ¼-modules, we get a linear basis +z , 2 , z , of Z such that P w(z )"s (w)z for all w3 ¼, G G G where s ’s are irreducible characters of ¼. G Using (5.10), we have (u?G z )(u?H z )"aKG\LG (u?H)u?G>?H (z z )"aKG\LG (u?H)u?H (u?G w (z ))z G H G H G H G "aKG\LG (u?H)u?H (w (z )w\ (u?G))z G H H G "aKG\LG (u?H)u?H
s (w )z a\KH\LH(u?G)u?G z H G H G
"aKG\LG (u?H)aLH\KH (u?G)s (w )(u?H z )(u?G z ). H G H G (3) follows from (AS\\T)(Z K L I )"Z \L 3 ?
. ?
\K \I>\?\
3
)
For a fixed v3 9, let & " : 1AS\\T2 be the cyclic subgroup of G¸(H) generated by the T anti-automorphism AS\\T. By Proposition 5.3(3), Z(u) is a & -module and every element T of & acts on Z(u) as either an algebra automorphism or an anti-automorphism. Now we T
RNote that w may be equal to w even for iOj. G H
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define ¹ : 9 ;9 P(9/"º"9), u : & PAut(¼), * : GP4 (9/"º"9) and t : GP(9/"º"9) by T 1!(!1)?@ ¹(a, b) " : , 2 u((AS\\T)R)(AKBLºI) " : A\R> K>\R\ LB\R> L>\R\ Kº\R I, *((AS\\T)R) " : (!1)R, (!1)R!1 , t((AS\\T)R) " : 2 where t, m, n, k 3 9 and a, b3 9 . It is easy to check that all assumptions in Proposition 4.3 are satisfied. Hence, by Proposition 4.3, we know that the action (AS\\T)R ° (AKBLºI, a) " : (u((AS\\T)R)(AKBLºI)º2? ?R 1\\TR,a) "(A\R> K>\R\ LB\R> L>\R\ K ;º\R I>\R\ \\? , a)
(5.11)
is an automorphism of the central map-product of ¼ by 9 determined by º and ¹. From (5.11), we can rewrite (5.9) as (AS\\T)R(Z
)"Z \\T R K L I K L3I ? 1 ° 3 ?
(5.12)
for t, m, n, k39 and a39 . Set ZM (u) " : ZM , U ? ZM " : Z and U ? U ?
& ° (w, a)"(w, a). T
satisfy x ° (w , a )"(w , a ) for i"1, 2 Then ZM (u) is a subalgebra of Z(u). In fact, if Z G G G G UG ?G and any x 3& , then, by (5.11), T x ° [(w , a ) * (w , a )]"[x ° (w , a )] * [x ° (w , a )]"(w , a ) * (w , a ), which means that ZM ZM "Z Z U ? U ? U ? U ? "ZM -ZM (u) -Z U ? * U ? U ? * U ? by Proposition 5.3(1). Definition 5.1. The subalgebra ZM " : ZM (u) defined above is called the nice subalgebra of H determined by the group-like element u. It is clear that the nice algebra ZM is a & -module and has the properties described by T Proposition 5.3(1) and (2). Since every element of the group & acts on ZM either as an T algebra automorphism or as an anti-automorphism, the set ZM &T " : +z 3 ZM "x(z)"z for x3 & , T
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is a Jordan subalgebra ZM >. ZM &T is called the Jordan algebra of invariants of & and an T element of ZM &T is called an invariant of & . It is the Jordan algebra of invariants of & that T T plays a key role to establish the connection between invariant theory and Kauffman’s Knot invariants. In fact, we will prove in next section that determining all traces on a class of quantum Hopf algebra is equivalent to determining the ‘‘homogeneous’’ subspace ZM &T5ZM of ZM &T.
6. TRACES AND INVARIANT THEORY FOR NON-COMMUTATIVE ALGEBRAS
We have seen in Section 2 that how a Kauffman’s knot invariant is constructed from a trace on a quantum algebra. In this section, we will prove that the problem of determining al traces on a class of quantum Hopf algebras can be reduced to the problem of finding invariants of a group. Let (H, S, R, G) be a quantum algebra. We are interested in the kind of functional p 3H* with the properties: p(xy)"p(ST(y)x) for x, y 3 H p(x)"p(GTS(x)
for x3 H
(6.1) (6.2)
where v is an integer. The set T (H*) " : +p 3 H*"p satisfies both (6.1) and (6.2), T forms a subspace of H*. We define T (H*) as the mth trace space for H. In particular T T (H*) is called the trace space for H, which consists of all traces on H. PROPOSITION 6.1. ¸et (H, S, R, G) be a quantum algebra. ¹hen the map ' : T (H*) g T T T (H*) given by ' : p >(p¤GT) for p3 T (H*) T T is a k-linear isomorphism. )
Proof. Use (6.1) and (6.2).
Definition 6.1. A quantum algebra (H, S, R, G) is said to be a quantum Hopf algebra if H is Hopf algebra with the antipode S. A quantum Hopf algebra is denoted by (H, *, e, S, R, G), where (H, *, e, S) gives the Hopf algebra structure on H and (H, S, R, G) gives the quantum algebra structure on H. An interesting behaviour of the trace space T (H*) for quantum Hopf algebra H is that T (H*) is a Jordan subalgebra of (H*)>. See [8] for details. Let H be a finite-dimensional Hopf algebra. By Proposition 3.2, we have a k-linear isomorphism ( : H* g H defined by ( : (j¤h) > h for h 3 H where j is a non-zero right integral for H*. If H is a finite-dimensional quantum Hopf algebra, we set r (H) " : ((¹ (H*)). T T
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The vector space r (H) is called the vth right counterpart of the trace space, which plays an T important role in the study of the trace space. PROPOSITION 6.3. ¸et (H, S, R, G) be a finite-dimensional quantum Hopf algebra. ¹hen r (H)"+GzG\T"(AS\\T)(z)"a(G)>Tzg\G and z3 Z ,. T Proof. Suppose that p"j¤h for some h 3 H, where j is a non-zero right integral for H*. Then for any x 3 H, by Proposition 3.2(2) and (3), we have p(GTS(x))"(j¤h)(GTS(x))"j(hGTS(x)) "(jS)(xG\TS\(h))"j(gxG\TS\(h)) "j(S([G\TS\(h)]¤a)gx)"(x-j)(S([G\TS\(h)]¤a)g) or p(GTS(x))"(x-j)(S([G\TS\(h)]¤a)g) for x3H.
(6.3)
p(x)"(x-j)(h) for x 3H.
(6.4)
It is clear that
It follows from (6.3) and (6.4) that p satisfies (6.2) if and only if (x-j)(S([G\TS\(h)]¤a)g)"(x-j)(h) for x3H. This prove that j¤h satisfies (6.2) if and only if hg\"(SA)[G\TS\(h)].
(6.5)
Similarly, one can prove that j¤h satisfies (6.1) if and only if h"GzG\T for some z3A . Replacing h by GzG\T in (6.5), we get (AS\\T)(z)"a(G)>Tzg\G. ) From [12], we know that the one-dimensional ideals of H* invariant under the antipode of H* are in one-to-one correspondance with the set +h3G(h)"h"g,. If we use a grouplike element gN with gN "g, then the description of r (H) can be rewritten as follows. T PROPOSITION 6.4. ¸et (H, S, R, G) be a finite-dimensional quantum Hopf algebra. If G, gN 3G(H) and gN "g, then r (H)"+GzG\T"(AS\\T)(gN \Gz)"a(G)>T(gN \Gz) and z 3 Z , T Proof. By Proposition 6.3, we have (AS\\T)(z)"a(G)>TzgN \G. Hence, using Lemma 5.2, we get a(G)>TzgN \G"(AS\\T)(z)G\gN "(AS\\T)(z)a(G\)\A(G\)a(gN )\A(gN ) "a(GgN \)A(S\\T(z)G\gN )"a(GgN \)A(S\\T(gN \Gz)) or (AS\\T)(gN \Gz)"a(GTgN )z(gN \G)"a(GTgN )A(gN \G)z "a(GTg )a(gN \G) (gN \Gz)"a(G)>T(gN \Gz). This proves Proposition 6.4.
)
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Let u " : gN \G. Then, by Proposition 3.2(4), for h 3 H, º(h)"g\GhG\g"g\S(h)g"a-h¤a\"(A\B\)(h) so º"A\B\ and ¼ " : 1A, B, º2"1A, B2"+AKBK"m, n3 9,. The nice subalgebra ZM (gN \G) of H determined by u " : gN \G is ZM (gN \G)" ZM K L ? K L9Z 9 ?Z where ZM K L " : +(gN \G)?z"zx"(AKBL)(x)z for all x 3 H, ? and (AKBL, a) is an element of the central map-product of ¼ by 9 with the following properties: : ZM \\T R ° K L (AS\\T)R(ZM K L ) " 1 ? ?
B
(!1)R#1 (!1)R!1 (!1)?!1 m# n# 2 2 2
;(AS\\T)R ° (AKBL, a) " : A
, a
(!1)R#1 (!1)R!1 (!1)?#1 n# m# 2 2 2
(AS\\T)R ° (AKBL, a)"(AKBL, a) for v, t, m, n3 9 and a39 . It is obvious that ZM -ZM (gN \G). By Proposition 6.4, we get the main result of this paper. PROPOSITION 6.5. ¸et (H, S, R, G) be a finite-dimensional quantum Hopf algebra. ¸et g be the distinguished grouplike element of H. If G 3 G(H) and g"gN for some gN 3 G(H), then r (H)"G(ZM &\5ZM )G, \ where ZM " : ZM (gN \G) is the nice subalgebra of H determined by the grouplike element gN \G, & is the cyclic group generated by the anti-automorphism AS, ZM &\ (-ZM ) is the Jordan \ algebra of invariants of the group & and ZM is the ‘‘homogeneous component’’ of ZM with \ ‘‘degree’’ (A, 1). ) Let (H, S, R, G) be a finite-dimensional quantum Hopf algebra. If H is unimodular, then a"e, ¼"1, & "1S2 and the nice subalgebra ZM " : ZM (gN \G) of H becomes a com\ mutative algebra: ZM "Z(H)#(gN \G)Z(H) where Z(H) is the center of H, gN "g and gN 3G(H). Suppose that Z(H)"k[z , 2 , z ], then P ZM "k[gN \G, z , 2 , z ]. Using Proposition 3.1, the algebra ZM &\ of invariants of & is P \ generated over k by SR((gN \G)? z? 2 z?P), P RZ9
a , a , 2 , a *0 and a #a 2#a )"S". P P
This proves that in the case that H is unimodular, the classical invariant theory is enough to determine the traces on H. If H is not unimodular, then aOe, ¼O1 and the nice subalgebra ZM (gN \G) is a non-commutative algebra. In this case, we need to develop the invariant theory for non-commutative algebras to determine thr traces on H.
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We finish this paper by restating the following fact: one of the most evident differences between the classical invariant theory and the invariant theory for non-commutative algebras is that if a group G acts on a non-commutative algebra R, then there maybe exists a normal subgroup N of G such that [G : N]"2 and N consists of all of the elements in G which act on the algebra R as automorphisms, hence, the set of invariants of G is generally a Jordan subalgebra of R> instead of an associative subalgebra of R. It is worthy of studying whether or not the new behaviour of invariant theory for non-commutative algebras will bring some interesting development to invariant theory. Acknowledgements—I would like to thank Prof. J. B. Carrell, Prof. R. V. Moody and Prof. D. Rolfsen for their support and useful conversations.
REFERENCES 1. Dieudonne, J. A. and Carrell, J. B., Invariant ¹heory Old and New, Academic Press, New York, 1971. 2. Doubilet, P. and Rota, G.-C., Skew-symmetric invariant theory. Advances in Mathematics, 1976, 21, 196—201. 3. Drinfeld, V. G., On almost cocommutative Hopf algebras. ¸eningrad Mathematical Journal, 1990, 1(2), 321—342. 4. Kauffmaan, L. H., Gauss codes, quantum groups and ribbon Hopf algebras. Reviews in Mathematical Physics, 1993, 5(4), 735—773. 5. Kauffmaan, L.H., Hopf algebras and invariants of 3-manifolds. Journal of Pure and Applied Algebra, 1995, 100, 73—92. 6. Kauffmaan, L. H. and Radford, D. E., Invariants of 3-manifolds derived from finite dimensional Hopf algebras. Journal of Knot ¹heory and its Ramifications, 1995, 4(1), 131—162. 7. Larson, R. G. and Sweedler, M. E., An associative orthogonal bilinear form for Hopf algebras. American Journal of Mathematics, 1969, 91, 75—94. 8. Liu, K., The trace space and Kauffman’s knot invariants. ¹ransactions of AMS (to appear). 9. Nichols, W. D., Cosemisimple Hopf algebras. In Advances in Hopf Algebras, eds. J. Bergen and S. Montgomery. 1994, pp. 135—151. 10. Radford, D. E., On Kauffman’s knot invariants arising from finite-dimensional Hopf algebras. In Advances in Hopf Algebras, eds. J. Bergen and S. Montgomery. 1994, pp. 205—266. 11. Radford, D. E., The trace function and Hopf algebras. Journal of Algebra, 1994, 163, 583—622. 12. Radford, D. E., The order of the antipode of a finite-dimensional Hopf algebra is finite. American Journal of Mathematics, 1976, 98, 333—355. 13. Richman, D., Explicit generators of invariants of finite groups. Advances in Mathematics, 1996, 124(1), 49—76. 14. Rota, G.-C. and Sturmfels, B., Introduction to invariant theory in superalgebras. in Invariant ¹heory and ¹ableaux, ed. D. Stanton. 1989, pp. 1—35. 15. Sloane, N. J. A., Error-correcting codes and invariant theory: New applications of a nineteenth-century technique. American Mathematical Monthly, 1977, 84, 82—107. 16. Stanley, R. P., Invariants of finite groups and their applications to combinatorics. Bulletin of American Mathematical Society, 1979, 1(3), 475—511. 17. Sweedler, M. E., Hopf Algebra. Mathematical Lecture Notes Series, Benjamin, New York, 1969.
Department of Mathematics The University of British Columbia Vancouver, BC, Canada V6T 1Z2
PII: S0040-9383(98)00014-7
INVARIANT THEORY AND THE INVARIANTS OF LOW-DIMENSIONAL TOPOLOGY KEQIN LIU (Received 27 December 1996; in revised form 2 December 1997; accepted 20 January 1998)
1. INTRODUCTION
The aim of this paper is to disclose the connection between invariant theory and the invariants of low-dimensional topology. The bridge which connects the two subjects is based upon the amazing connection between the deep algebraic structures of Hopf algebras and Kauffman—Radford—Hennings invariants of 3-manifolds, where Kauffman—Radford— Hennings invariants of 3-manifolds are a new class of 3-manifold invariants constructed by employing the right integrals on certain Hopf algebras in [4—6]. Since Kauffman—Radford— Hennings invariants of 3-manifolds are just some scalar multiples of Kauffman’s knot invariants and every trace on a quantum algebra produces a Kauffman’s knot invariant, so the study of traces on quantum algebras plays a key role in the study of Kauffman—Radford—Hennings invariants of 3-manifolds. In this paper, we will prove that determining the traces on a quantum Hopf algebra is equivalent to determining a ‘‘homogeneous’’ part of the algebra of invariants of a group acting on an associative algebra ZM . This fact not only joins the classical invariant theory to Kaufman’s knot invariants in the case that ZM is commutative, but also shows us the necessity of considering the invariants theory for non-commutative algebras in the case that ZM is not commutative. The invariant theory for some non-commutative algebras like exterior algebras and superalgebras has been studied in [2, 14] for different motivations. The connection between invariant theory and the invariants of low-dimensional topology provides another motivation of studying invariant theory for non-commutative algebras. Hopefully, invariant theory can eventually arise again from the realm of non-commutative algebras. This paper is organized as follows. In Section 2 we sketch the construction of Kauffman’s knot invariants and exhibit the connection between Kauffman’s knot invariants and quantum algebras with a trace. In Section 3 we recall some facts in classical invariant theory and some properties about integrals on finite-dimensional Hopf algebras. In Section 4 we generalize semi-products in group theory and introduce the notion of the map-product of two groups. The notion of map-products, which is of independent interest, is the essential part in establishing the connection between invariant theory and the study of the traces on quantum Hopf algebras. In Section 5 we construct the nice algebra ZM " : ZM (u) from a quantum Hopf algebra H and a group-like element u3H with the property that the elements of a group G act on ZM as either algebra automorphisms or anti-automorphisms. Therefore, the set ZM % " : +z3 ZM "x(z)"z for all x3G, forms of Jordan subalgebra of ZM >, where ZM > is the Jordan algebra obtained from the associative algebra ZM . ZM % is called the Jordan algebra of invariants of the group G. In 763
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Section 6 we prove that determining the traces on the quantum algebra H is the same as determining a ‘‘homogeneous’’ part of the Jordan algebra ZM % of invariants of the group G.
2. KAUFFMAN’S KNOT INVARIANTS
In this section we explain the topological background of studying quantum algebras. Since it is not necessary to use invariants that depend on link orientation to define 3-manifold invariants, we consider only the invariants for unoriented links. Kauffman proves that an invariant of regular isotopy of unoriented knots can be constructed from a quantum algebra with a trace, where quantum algebras are a class of associative algebras introduced by Kauffman in [4]. Definition 2.1. An associative algebra H over a commutative ring k is called a quantum algebra if there exist S3Hom (H, H), R3 H H and G 3 H such that I (1) S(xy)"S(yx) for x, y3 H, (2) R R R "R R R , (3) R\"(S 1)(R)"(1 S\)(R), (4) S(G)"G\ and S(x)"GxG\ for x3 H, where R " : R 13 H H H, R :"(1 p)(R ), R " : (p 1)(R ) and p is the twist automorphism of H H given by p : x y > y x for x, y 3 H. The quantum algebra H depending on S, R and G is denoted by (H, S, R, G). A useful corollary of (3) in Definition 2.1 is the equations (SK SK)(R$)"R$ for m3 9
(2.1)
Definition 2.2. Let (H, S, R, G) be a quantum algebra. A functional tr 3 H* " : Hom (H, k) is said to be a trace on H if it satisfies the following properties: I (1) tr(xy)"tr(yx) for x, y3 H, (2) tr(x)"tr(S(x)) for x 3 H. There are four basic building blocks in a knot diagram with respect to the vertical direction (of the paper). They are minima, maxima and two types of crossing:
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For example, the trefoil knot has two maxima, two minima and three crossings:
The four kinds of basic building blocks of the trefoil knot can be seen clearly if we use some vertical line segments in its diagram as follows:
This kind of description of a knot diagram is convenient in the construction of Kauffman’s knot invariants. Let (H, S, R, G) be a quantum algebra with a trace tr3 H*, where R" e f and G G G e , f 3 H. We call e f the general term of R with the sum index i. By Definition 2.1 (3), G G G G both S(e ) f and e S\( f ) is the general term of R\. Given a knot diagram K, we G G G G decorate the crossings of K by using the general term of R or R\ in the following ways:
In order to indicate different crossings, we use different sum index for the general term of R or R\. For example, the diagram of trefoil knot can be decorated as
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where we have used three different indices i, j and k to indicate the three crossings of the trefoil knot. We think of the knot diagram K as a closed loop of wire around which beads can be moved, and regard elements of the quantum algebra H appearing in the knot diagram K as beads. Choose a point on a vertical line segment in the diagram which is not part of a crossing. Moving upward from that point, slide all beads around the diagram into the vertical segment. As a bead moves across maxima or minima its label will change according to the following rule:
The final juxtaposition of labelled beads at the chosen vertical line segment gives rise to a product in H with finitely many sum indices (down to up is interpreted as multiplication left to right). Using the product as the general term, we get a sum w in H, where the sum is taken over all different sum indices appearing in the product. Let d be the Whitney degree of the diagram K that is obtained by traversing K upward from the chosen vertical line segment, where the beads have been concentrated. Then Kauffman’s knot invariant for K is defined by ¹R(K) :"tr(GBw). For example, if we move all beads in Fig. 1 from the point indicated by ;, then
Fig. 1.
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and d"!2. The sum w in H with the general term: S(e ) S( f )S(e )S( f )S(e ) f is G H I G H I w" S(e )S( f )S(e )S( f )S(e ) f . G H I G H I G H I The Kauffman’s knot invariant for the trefoil knot is ¹R(K)" tr (G\S(e )S( f )S(e )S( f )S(e ) f ) G H I G H I G H I which can be rewritten as ¹R(K)" tr(G\S( f )S(e )S( f )e f e ) G H I G H I G H I by (2.1). Note that the value of ¹R(K) is independent of the choice of vertical segment along the diagram for K at which the beads have been concentrated.
3. PRELIMINARIES ABOUT INVARIANT THEORY AND HOPF ALGEBRAS
For convenience, from this section we will assume that algebras and coalgebras are over a field k with char(k)O2. First, let us recall some basic facts in classical invariant theory. Let R"k[a , 2 , a ] be L a finitely generated commutative algebra over k, where the a ’s may be not algebraically G independent over k. Suppose that a group G acts on R by algebra automorphisms. An element f of R is called an invariant of G if x( f )"f for all x 3 G. The set consisting of all invariants of G is called the ring of invariants and denoted by R%, i.e. R%"+ f 3R"x( f )"f for all x3G,. Clearly, R% is a subalgebra of R. The basic object of study of invariant theory is the ring of invariants R%. Many interesting theorems have been found in order to describe R%. For our purpose, we need a theorem proved by Richman in [13]. PROPOSITION 3.1. Suppose that a finite solvable group G acts on a finitely generated commutative k-algebra R"k[a , 2 , a ]. If the order "G" of G is not divisible by the L characteristic of the field k, then the ring of invariants R% is generated over k by x(aG aG 2 aGL), L V Z% where i , 2 , i *0 and i #2#i )"G". L L Proof. Proposition 5 in [13].
)
Now let us recall some properties about integrals on finite-dimensional Hopf algebras. Let (H, *, e, S) be a finite-dimensional Hopf algebra with comultiplication *, counit e and antipode S. Let H* " : Hom (H, k) be the linear dual of H. Then H* is also a Hopf algebra. I We use G(H) to denote the set consisting of all groupslike elements of H. Then G(H*)"Alg (H, k); that is, G(H*) consists of all algebra homomorphisms from the algebra I H to the algebra k.
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Let " be a non-zero left integral for H. Since the ideal of left integrals for H is one-dimensional. Hence, there exists a unique a3 G(H*) such that "h"a(h)" for h3 H.
(3.1)
Similarly, let j be a non-zero right integral for H*, then exists a unique g3 G(H) such that pj"p(g)j for p3 H*.
(3.2)
Definition 3.1. Let " be a non-zero left integral for H and j be a non-zero right integral for H*. Then the unique group-like element a satisfying (3.1) is called the distinguished group-like element of H*, and the unique group-like element g satisfying (3.2) is called the distinguished group-like element of H. H* is a H-bimodule under the following actions: (h-p)(x) " : p(xh),
(p ¤h)(x) " : p(hx),
and H is a H*-bimodule under the following actions: p-h " : h p(h ), h¤p " : p(h )h , F F where h, x3H and p3 H*. PROPOSITION 3.2. ¸et (H, *, e, S) be a finite-dimensional Hopf algebra and j a non-zero right integral for H*. ¸et g be the distinguished grouplike element of H and let a be the distinguished grouplike element of H*. (1) (H*, ¤) is a free right H-module with basis j. (H*, -) is a free left H-module with basis j. (2) j(xy)"j(S(y¤a)x) for x, y3H. (3) j(S(x))"j(gx) for x 3 H. (4) Radford’s formula holds: S(x)"g(a-x¤a\)g\ for x 3H. (5) H is unimodular if and only if a"e. (6) H* is unimodular if and only if g"1. Proof. The proof of (1)—(4) can be found in [11]. (5) and (6) are clear.
)
4. MAP-PRODUCTS OF TWO GROUPS
Let K and ¼ be two groups. We use map(¼, ¼) to denote the set consisting of all maps from ¼ to ¼. Definition 4.1. A map u : K;KPmap(¼, ¼) is called a group-producing map if u satisfies the following conditions: (1) For k , k , k 3K and w , w 3 ¼, u(k k , k )[u(k , k )(w )w ]"u(k , k k )(w )u(k , k )(w ).
(4.1)
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(2) There exists an element w of ¼ such that u(1 , k)(w )"1 , u(k, 1 )(w)"ww\ for k3 K and w 3¼. ) 5 )
(4.2)
PROPOSITION 4.1. Given two groups K and ¼ and a map u : K;KPmap(¼, ¼). ¹hen the following are equivalent: (1) u is a group-producing map. (2) ¼;K is a group under the product (w , k )(w , k ) " : (u(k , k )(w )w , k k ) for w 3¼ and k 3 K, G G
(4.3)
where (w , 1 ) is the identity and (w, k)\"(u(k, k\)(w)\w , k\) for w3 ¼ and k 3 K. ) )
Proof. A direct calculation.
Definition 4.2. The group defined by (4.3) is called the map-product of ¼ by K via the map u. Example. Let ¼ and K be two groups. Let t : KPAut(¼) be a group homomorphism. Define a map u : K;KPmap(¼, ¼) by u(k , k )(w) " : wRI for w 3 ¼ and k , k 3K. Then it is clear that u is a group-producing map, and the map-product of ¼ by K via the map u is the semi-direct product of ¼ by K. Remark Like the semi-direct product, the map-product of ¼ by K contains a normal subgroup which is isomorphic to the group ¼. Another important special case of the map-product of ¼ by K is from some central elements of the group ¼. Let x be an element of a group. We set
"x" " :
the order of x if the order of x is finite 0 if the order of x is infinite.
PROPOSITION 4.2. Given two groups ¼ and K and a map ¹ : K;KP9/"h"9, where h is in the center of the group ¼. ¹hen ¼;K becomes a group under the product: (w , k )(w , k ) " : (w w h2I I, k k )
(4.4)
if and only if the map ¹ satisfies the conditions: ¹(k , k )#¹(k k , k ),¹(k , k k )
(mod"h"),
¹(1 , k),¹(k, 1 ) (mod"h"), ) ) where w , w 3¼ and k, k , k , k 3 K. Proof. Define a map u : K;KPmap(¼, ¼) by u(k , k )(w) " : wh2I I
for k , k 3 K and w3 ¼.
(4.5) (4.6)
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Then it is easy to check that u is a group-producing map if and only if ¹ satisfies (4.5) and (4.6). So Proposition 4.2 follows from Proposition 4.1. ) Definition 4.3. The map-product of ¼ by K defined in Proposition 4.2 is called the central map-product of ¼ by K determined by h and ¹. We finish this section by giving a sufficient condition for the claim that a group G can act on the central map-product of ¼ by K determined by h and ¹. PROPOSITION 4.3. Given three groups G, ¼ and K and a map ¹ : K;KP9/"h"9, where h is in the center of the group ¼ and ¹ satisfies (4.5) and (4.6). ¸et u : GPAut(¼) be a group homomorphism such that u(x)(h)"h V for some group homomorphism * : GP4(9/"h"9), where 4,(9/"h"9) is the unit group of (9/"h"9). (1) If a map t : GP9/"h"9 has the property t(xy),t(x)#*(x)t(y) (mod"h")
for x, y, 3G,
then the group G can act on the central map-product of ¼ by K determined by h and ¹ in the following way: x ° (w, k)"(u(x)(w)h2I IR V, k) for x3 G, w 3 ¼ and k3 K.
(4.7)
(2) If t satisfies both (4.7) and ¹(k , k )(*(x)!1)"(¹(k , k )#¹(k , k )!¹(k k , k k ))t(x), where k , k 3K and x 3G, then the action defined by (4.7) is an automorphism of the central map-product of ¼ by K determined by h and ¹.
5. THE NICE SUBALGEBRAS OF HOPF ALGEBRAS
In this section we assume that (H, S, R, G) is a finite-dimensional Hopf algebra over a field k, where char(k)"0. We will construct the nice subalgebras from H and grouplike elements of H, and prove that a cyclic subgroup of G¸(H) acts on the nice subalgebras naturally as either algebra automorphisms or anti-automorphisms. LEMMA 5.1. ¸et g3G(H*) and u3G(H) (1) Both g- and g¤ is an algebra automorphism of H. (2) For any h 3 H, we have g-(uhu\)"u(g-h)u\, (uhu\)¤g"u(h¤g)u\ S(g-h)"g-S(h), S(h¤g)"S(h)¤g.
(5.1) (5.2)
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Proof. (1) For x, y 3 H we have (xy)¤g" g(x y )x y V W
" g(x )x g(y )y "(x¤g)(y¤g). V W Similarly, one can check that g- is an algebra automorphism of H. (2) (5.1) follows from (1) and (5.2) is clear.
)
Let u 3 G(H) and a be the distinguished group-like element of H*. Define A, B, º 3 Aut(H) by A(h) " : h¤a, B(h) " : a\-h, º(h) " : uhu\ where h 3H. We use ¼ to denote the subgroup of Aut(H) generated by A, B and º. It is clear that ¼ is an abelian group and ¼" : +AKBLºI"m, n, k 3 9,. LEMMA 5.2. For any m, n, k 3 9, we have (AKBL)(h)"aK\L(h)h for h3G(H). S\AKBLºIS"ALBKºI.
(5.3) (5.4)
Proof. (5.3) is clear. (5.4) follows from the following two identities: S\AKS"BK for m 3 9
(5.5)
S\BLS"AL for n39.
(5.6) )
For w3¼ and a3 9 " : +0, 1,, we set " : +u?z"zx"w(x)z for x3 H,, Z U ? Z" Z for a 3 9 , ? U ? UZ5 Z(u) " : Z #Z . It is easy to check that 1!(!1)??Y ¹(a, a ) " : for a, a 3 9 2 is a map form 9 ;9 to 9/"º"9 satisfying both (4.5) and (4.6). By Proposition 4.2, we have the central map-product of ¼ by 9 determined by º and ¹ whose group operation * is given by (w, a) * (w , a )"(ww º2? ?Y, a#a ) for w, w 3 ¼ and a, a 39 . PROPOSITION 5.3. ¹he associative algebra Z(u) has the following properties: (1) Z(u) is ‘‘graded’’ by the central map-product of ¼ by 9 determined by º and ¹ in the sense: Z Z -Z "Z , (5.7) U ? UY ?Y U ? * UY ?Y UUY32? ?Y ?>?Y where w, w 3 ¼ and a, a 3 9 .
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(2) ¸et r"dim Z . ¹here exists a linear basis +z , 2 , z , of Z (-8 Z ) and P U Z 5 U r irreducible character s , 2 , s of the abelian group ¼ such that Z(u) is spanned P linearly by +u?z "a3+0, 1,, 1)i)r, and G (u?G z )(u?H z )"aKG\LG (u?H)aLH\KH (u?G)s (w )(u?H z )(u?G z ), G H H G H G where 1)i, j)r, a 39 , z 3Z - and w "AKGBLGºIG 3¼ for some m , n , k 39. G G UG G G G G (3) For v, t, m, n, k39 and a3 9 , (AS\\T)R(Z K L I )"Z \R> K>\R\ L 3 ?
\R> L>\R\ K \RI>\R\ \\?
3
. ? (5.9)
Proof. (1) For convenience, we use z to indicate that z is an element of Z . So U U U z x"w(x)z for all x3 H and z z 3 Z . By (5.3), we get U U U UY UUY (u?z )(u?Yz )"aK\L(u?Y)u?>?Yz z , U UY U UY
(5.10)
where a, a 3 9 , w, w 3 ¼ and w"AKBLºI for some m, n, k39. It follows from (5.10) that (5.7) is true if one of a and a is 0. In order to prove that (5.7) is also true for a"a "1, it suffices to prove that uz z 3 Z , which holds by the following: U UY UUY3 (uz z )x"u(z z x)"uww (x)z z "uww (x)u\uz z U UY U UY U UY U UY "(ºww )(x)(uz z ) for x 3H. U UY (2) Since Z is ¼-invariant and ¼ is finite, Z is a complete reducible ¼-module U U for every w 3¼. This implies that there exists a ¼-submodule Z of Z such that U U Z " Z . Expressing every non-zero Z as a direct sum of one-dimensional U Z 5 U U irreducible ¼-modules, we get a linear basis +z , 2 , z , of Z such that P w(z )"s (w)z for all w3 ¼, G G G where s ’s are irreducible characters of ¼. G Using (5.10), we have (u?G z )(u?H z )"aKG\LG (u?H)u?G>?H (z z )"aKG\LG (u?H)u?H (u?G w (z ))z G H G H G H G "aKG\LG (u?H)u?H (w (z )w\ (u?G))z G H H G "aKG\LG (u?H)u?H
s (w )z a\KH\LH(u?G)u?G z H G H G
"aKG\LG (u?H)aLH\KH (u?G)s (w )(u?H z )(u?G z ). H G H G (3) follows from (AS\\T)(Z K L I )"Z \L 3 ?
. ?
\K \I>\?\
3
)
For a fixed v3 9, let & " : 1AS\\T2 be the cyclic subgroup of G¸(H) generated by the T anti-automorphism AS\\T. By Proposition 5.3(3), Z(u) is a & -module and every element T of & acts on Z(u) as either an algebra automorphism or an anti-automorphism. Now we T
RNote that w may be equal to w even for iOj. G H
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define ¹ : 9 ;9 P(9/"º"9), u : & PAut(¼), * : GP4 (9/"º"9) and t : GP(9/"º"9) by T 1!(!1)?@ ¹(a, b) " : , 2 u((AS\\T)R)(AKBLºI) " : A\R> K>\R\ LB\R> L>\R\ Kº\R I, *((AS\\T)R) " : (!1)R, (!1)R!1 , t((AS\\T)R) " : 2 where t, m, n, k 3 9 and a, b3 9 . It is easy to check that all assumptions in Proposition 4.3 are satisfied. Hence, by Proposition 4.3, we know that the action (AS\\T)R ° (AKBLºI, a) " : (u((AS\\T)R)(AKBLºI)º2? ?R 1\\TR,a) "(A\R> K>\R\ LB\R> L>\R\ K ;º\R I>\R\ \\? , a)
(5.11)
is an automorphism of the central map-product of ¼ by 9 determined by º and ¹. From (5.11), we can rewrite (5.9) as (AS\\T)R(Z
)"Z \\T R K L I K L3I ? 1 ° 3 ?
(5.12)
for t, m, n, k39 and a39 . Set ZM (u) " : ZM , U ? ZM " : Z and U ? U ?
& ° (w, a)"(w, a). T
satisfy x ° (w , a )"(w , a ) for i"1, 2 Then ZM (u) is a subalgebra of Z(u). In fact, if Z G G G G UG ?G and any x 3& , then, by (5.11), T x ° [(w , a ) * (w , a )]"[x ° (w , a )] * [x ° (w , a )]"(w , a ) * (w , a ), which means that ZM ZM "Z Z U ? U ? U ? U ? "ZM -ZM (u) -Z U ? * U ? U ? * U ? by Proposition 5.3(1). Definition 5.1. The subalgebra ZM " : ZM (u) defined above is called the nice subalgebra of H determined by the group-like element u. It is clear that the nice algebra ZM is a & -module and has the properties described by T Proposition 5.3(1) and (2). Since every element of the group & acts on ZM either as an T algebra automorphism or as an anti-automorphism, the set ZM &T " : +z 3 ZM "x(z)"z for x3 & , T
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is a Jordan subalgebra ZM >. ZM &T is called the Jordan algebra of invariants of & and an T element of ZM &T is called an invariant of & . It is the Jordan algebra of invariants of & that T T plays a key role to establish the connection between invariant theory and Kauffman’s Knot invariants. In fact, we will prove in next section that determining all traces on a class of quantum Hopf algebra is equivalent to determining the ‘‘homogeneous’’ subspace ZM &T5ZM of ZM &T.
6. TRACES AND INVARIANT THEORY FOR NON-COMMUTATIVE ALGEBRAS
We have seen in Section 2 that how a Kauffman’s knot invariant is constructed from a trace on a quantum algebra. In this section, we will prove that the problem of determining al traces on a class of quantum Hopf algebras can be reduced to the problem of finding invariants of a group. Let (H, S, R, G) be a quantum algebra. We are interested in the kind of functional p 3H* with the properties: p(xy)"p(ST(y)x) for x, y 3 H p(x)"p(GTS(x)
for x3 H
(6.1) (6.2)
where v is an integer. The set T (H*) " : +p 3 H*"p satisfies both (6.1) and (6.2), T forms a subspace of H*. We define T (H*) as the mth trace space for H. In particular T T (H*) is called the trace space for H, which consists of all traces on H. PROPOSITION 6.1. ¸et (H, S, R, G) be a quantum algebra. ¹hen the map ' : T (H*) g T T T (H*) given by ' : p >(p¤GT) for p3 T (H*) T T is a k-linear isomorphism. )
Proof. Use (6.1) and (6.2).
Definition 6.1. A quantum algebra (H, S, R, G) is said to be a quantum Hopf algebra if H is Hopf algebra with the antipode S. A quantum Hopf algebra is denoted by (H, *, e, S, R, G), where (H, *, e, S) gives the Hopf algebra structure on H and (H, S, R, G) gives the quantum algebra structure on H. An interesting behaviour of the trace space T (H*) for quantum Hopf algebra H is that T (H*) is a Jordan subalgebra of (H*)>. See [8] for details. Let H be a finite-dimensional Hopf algebra. By Proposition 3.2, we have a k-linear isomorphism ( : H* g H defined by ( : (j¤h) > h for h 3 H where j is a non-zero right integral for H*. If H is a finite-dimensional quantum Hopf algebra, we set r (H) " : ((¹ (H*)). T T
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The vector space r (H) is called the vth right counterpart of the trace space, which plays an T important role in the study of the trace space. PROPOSITION 6.3. ¸et (H, S, R, G) be a finite-dimensional quantum Hopf algebra. ¹hen r (H)"+GzG\T"(AS\\T)(z)"a(G)>Tzg\G and z3 Z ,. T Proof. Suppose that p"j¤h for some h 3 H, where j is a non-zero right integral for H*. Then for any x 3 H, by Proposition 3.2(2) and (3), we have p(GTS(x))"(j¤h)(GTS(x))"j(hGTS(x)) "(jS)(xG\TS\(h))"j(gxG\TS\(h)) "j(S([G\TS\(h)]¤a)gx)"(x-j)(S([G\TS\(h)]¤a)g) or p(GTS(x))"(x-j)(S([G\TS\(h)]¤a)g) for x3H.
(6.3)
p(x)"(x-j)(h) for x 3H.
(6.4)
It is clear that
It follows from (6.3) and (6.4) that p satisfies (6.2) if and only if (x-j)(S([G\TS\(h)]¤a)g)"(x-j)(h) for x3H. This prove that j¤h satisfies (6.2) if and only if hg\"(SA)[G\TS\(h)].
(6.5)
Similarly, one can prove that j¤h satisfies (6.1) if and only if h"GzG\T for some z3A . Replacing h by GzG\T in (6.5), we get (AS\\T)(z)"a(G)>Tzg\G. ) From [12], we know that the one-dimensional ideals of H* invariant under the antipode of H* are in one-to-one correspondance with the set +h3G(h)"h"g,. If we use a grouplike element gN with gN "g, then the description of r (H) can be rewritten as follows. T PROPOSITION 6.4. ¸et (H, S, R, G) be a finite-dimensional quantum Hopf algebra. If G, gN 3G(H) and gN "g, then r (H)"+GzG\T"(AS\\T)(gN \Gz)"a(G)>T(gN \Gz) and z 3 Z , T Proof. By Proposition 6.3, we have (AS\\T)(z)"a(G)>TzgN \G. Hence, using Lemma 5.2, we get a(G)>TzgN \G"(AS\\T)(z)G\gN "(AS\\T)(z)a(G\)\A(G\)a(gN )\A(gN ) "a(GgN \)A(S\\T(z)G\gN )"a(GgN \)A(S\\T(gN \Gz)) or (AS\\T)(gN \Gz)"a(GTgN )z(gN \G)"a(GTgN )A(gN \G)z "a(GTg )a(gN \G) (gN \Gz)"a(G)>T(gN \Gz). This proves Proposition 6.4.
)
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Let u " : gN \G. Then, by Proposition 3.2(4), for h 3 H, º(h)"g\GhG\g"g\S(h)g"a-h¤a\"(A\B\)(h) so º"A\B\ and ¼ " : 1A, B, º2"1A, B2"+AKBK"m, n3 9,. The nice subalgebra ZM (gN \G) of H determined by u " : gN \G is ZM (gN \G)" ZM K L ? K L9Z 9 ?Z where ZM K L " : +(gN \G)?z"zx"(AKBL)(x)z for all x 3 H, ? and (AKBL, a) is an element of the central map-product of ¼ by 9 with the following properties: : ZM \\T R ° K L (AS\\T)R(ZM K L ) " 1 ? ?
B
(!1)R#1 (!1)R!1 (!1)?!1 m# n# 2 2 2
;(AS\\T)R ° (AKBL, a) " : A
, a
(!1)R#1 (!1)R!1 (!1)?#1 n# m# 2 2 2
(AS\\T)R ° (AKBL, a)"(AKBL, a) for v, t, m, n3 9 and a39 . It is obvious that ZM -ZM (gN \G). By Proposition 6.4, we get the main result of this paper. PROPOSITION 6.5. ¸et (H, S, R, G) be a finite-dimensional quantum Hopf algebra. ¸et g be the distinguished grouplike element of H. If G 3 G(H) and g"gN for some gN 3 G(H), then r (H)"G(ZM &\5ZM )G, \ where ZM " : ZM (gN \G) is the nice subalgebra of H determined by the grouplike element gN \G, & is the cyclic group generated by the anti-automorphism AS, ZM &\ (-ZM ) is the Jordan \ algebra of invariants of the group & and ZM is the ‘‘homogeneous component’’ of ZM with \ ‘‘degree’’ (A, 1). ) Let (H, S, R, G) be a finite-dimensional quantum Hopf algebra. If H is unimodular, then a"e, ¼"1, & "1S2 and the nice subalgebra ZM " : ZM (gN \G) of H becomes a com\ mutative algebra: ZM "Z(H)#(gN \G)Z(H) where Z(H) is the center of H, gN "g and gN 3G(H). Suppose that Z(H)"k[z , 2 , z ], then P ZM "k[gN \G, z , 2 , z ]. Using Proposition 3.1, the algebra ZM &\ of invariants of & is P \ generated over k by SR((gN \G)? z? 2 z?P), P RZ9
a , a , 2 , a *0 and a #a 2#a )"S". P P
This proves that in the case that H is unimodular, the classical invariant theory is enough to determine the traces on H. If H is not unimodular, then aOe, ¼O1 and the nice subalgebra ZM (gN \G) is a non-commutative algebra. In this case, we need to develop the invariant theory for non-commutative algebras to determine thr traces on H.
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We finish this paper by restating the following fact: one of the most evident differences between the classical invariant theory and the invariant theory for non-commutative algebras is that if a group G acts on a non-commutative algebra R, then there maybe exists a normal subgroup N of G such that [G : N]"2 and N consists of all of the elements in G which act on the algebra R as automorphisms, hence, the set of invariants of G is generally a Jordan subalgebra of R> instead of an associative subalgebra of R. It is worthy of studying whether or not the new behaviour of invariant theory for non-commutative algebras will bring some interesting development to invariant theory. Acknowledgements—I would like to thank Prof. J. B. Carrell, Prof. R. V. Moody and Prof. D. Rolfsen for their support and useful conversations.
REFERENCES 1. Dieudonne, J. A. and Carrell, J. B., Invariant ¹heory Old and New, Academic Press, New York, 1971. 2. Doubilet, P. and Rota, G.-C., Skew-symmetric invariant theory. Advances in Mathematics, 1976, 21, 196—201. 3. Drinfeld, V. G., On almost cocommutative Hopf algebras. ¸eningrad Mathematical Journal, 1990, 1(2), 321—342. 4. Kauffmaan, L. H., Gauss codes, quantum groups and ribbon Hopf algebras. Reviews in Mathematical Physics, 1993, 5(4), 735—773. 5. Kauffmaan, L.H., Hopf algebras and invariants of 3-manifolds. Journal of Pure and Applied Algebra, 1995, 100, 73—92. 6. Kauffmaan, L. H. and Radford, D. E., Invariants of 3-manifolds derived from finite dimensional Hopf algebras. Journal of Knot ¹heory and its Ramifications, 1995, 4(1), 131—162. 7. Larson, R. G. and Sweedler, M. E., An associative orthogonal bilinear form for Hopf algebras. American Journal of Mathematics, 1969, 91, 75—94. 8. Liu, K., The trace space and Kauffman’s knot invariants. ¹ransactions of AMS (to appear). 9. Nichols, W. D., Cosemisimple Hopf algebras. In Advances in Hopf Algebras, eds. J. Bergen and S. Montgomery. 1994, pp. 135—151. 10. Radford, D. E., On Kauffman’s knot invariants arising from finite-dimensional Hopf algebras. In Advances in Hopf Algebras, eds. J. Bergen and S. Montgomery. 1994, pp. 205—266. 11. Radford, D. E., The trace function and Hopf algebras. Journal of Algebra, 1994, 163, 583—622. 12. Radford, D. E., The order of the antipode of a finite-dimensional Hopf algebra is finite. American Journal of Mathematics, 1976, 98, 333—355. 13. Richman, D., Explicit generators of invariants of finite groups. Advances in Mathematics, 1996, 124(1), 49—76. 14. Rota, G.-C. and Sturmfels, B., Introduction to invariant theory in superalgebras. in Invariant ¹heory and ¹ableaux, ed. D. Stanton. 1989, pp. 1—35. 15. Sloane, N. J. A., Error-correcting codes and invariant theory: New applications of a nineteenth-century technique. American Mathematical Monthly, 1977, 84, 82—107. 16. Stanley, R. P., Invariants of finite groups and their applications to combinatorics. Bulletin of American Mathematical Society, 1979, 1(3), 475—511. 17. Sweedler, M. E., Hopf Algebra. Mathematical Lecture Notes Series, Benjamin, New York, 1969.
Department of Mathematics The University of British Columbia Vancouver, BC, Canada V6T 1Z2