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Casimir invariants for Hopf algebras
Vol. 31 (1992)
REPORTS
ON MATHEMATICAL
CASIMIR INVARIANTS
PHYSICS
No. 1
FOR HOPF ALGEBRAS
J. R. LINKS and M. D. GOULD Department of Mathematics, University of Queensland, St. Lucia, QLD, Australia 4067 (Received May 14, 1991) A full set of invariants for an arbitrary quantum group is constructed which reduce to the Gel’fand invariants of the corresponding Lie algebra in the zero deformation limit q -+ 1. The eigenvalues of these invariants on irreducible representations are also determined as a simple function of the highest weight labels. Extensions to more general classes of Hopf algebra are also discussed.
1. Introduction
Recently, there has been renewed interest in the study of Hopf algebras because of their role in obtaining solutions to the Yang-Baxter equation, which arises in areas such as integrable lattice models [l], the quantum inverse scattering method [2] and the theory of knots and links [3]. In particular, it is the so called quasi-triangular Hopf algebras as defined by Drinfeld [4] that have received most attention since they admit a canonical element known as the universal (i.e. representation independent) R-matrix which automatically provides a solution to the Yang-Baxter equation. From the point of view of physical applications of quasi-triangular Hopf algebras, it is necessary to study their representation theory. This in turn leads us to a study of the centralizer of these algebras, as we expect that knowledge of the centralizer will prove to be useful in their general representation theory. In this paper we investigate some methods of constructing invariants for Hopf algebras. Quantum groups [4, 61 are examples of quasi-triangular Hopf algebras which are obtained by deforming the universal enveloping algebra of a simple Lie algebra through the introduction of a non-zero deformation parameter q. As q -+ 1, the defining relations of a quantum group reduce to those of a simple Lie algebra. The representation theory of quantum groups appears to be very similar to that of simple Lie algebras. In particular, it is now known [8] that when q is not a root of unity, all finite dimensional representations of quantum groups are completely reducible and the irreducible ones can be classified in terms of their highest weights. One of the main aims of this paper is to determine the analogues of the Gel’fand invariants for quantum groups. A set of invariants generating the centre of the quantum group, which do not give rise to the Gel’fand invariants of the corresponding Lie algebra as q + 1, has already been determined in general terms by Gould et al. [9] and Faddeev et al. [16]. In this paper we construct an alternative set of invariants (and their eigenvalues) which reduce [911
J. R. LINKS and M. D. GOULD
92
to the Gel’fand invariants as 4 -+ 1. We follow the general approach of ref. [9] although our construction is simpler and more direct; in particular, fully explicit results are obtained for the generalized Gel’fand invariants of U,(gl(n)). We expect that these invariants will be easier to use for both practical and theoretical considerations. In addition, we will determine the images of these generalized Gel’fand invariants under the action of the antipode, and determine characteristic polynomial identities for quantum groups which are generalizations of those obtained by Bracken and Green [13] for simple Lie algebras. These polynomial identities allow us to construct projection operators which play an important role in constructing solutions of the Yang-Baxter equation [4, 61. This paper begins with some general results on Hopf algebras and invariant bilinear forms. We will show that there is a particular class of Hopf algebras which always admit invariant bilinear forms. Such a form affords a unified construction of Casimir invariants, as we will see. All quasi-triangular Hopf algebras fall into this class so the theory is readily applicable to them. In this case however, the universal R-matrix enables an alternative construction of Casimir invariants. This general theory is then used to determine the generalized Gel’fand invariants and characteristic polynomial identities for quantum groups. As an example, we will consider U,(gl(n)) in detail. 2. Fundamentals Let A denote a Hopf algebra with co-product A : A --f A @ A, bijective antipode S : A -+ A and co-unit E : A + C, where C is the underlying complex field. We will not give the full defining relations of a Hopf algebra here but refer the reader to refs. [14, 151 for details. However, using the Sigma notation of Sweedler [14] we note the following important properties:
c
CL(l) @. . . @ ,(,G)),@+l)
@ . . . @ @+l)
(a)
=
c
u(l) @. . . 63 u(n)
=g
u(l) @. .
=g
a(l) @ . . . @ &)q&+‘$-&i+*)
~3 A(&))
@ . . . @ c&n-1) @ . . . gJ a(n+*),
(1)
(a)
c a(‘)@
. . . @A(S(di’))@.
. @~a(~-~) = c a(%. (a)
(a)
The adjoint representation under the action
. . cm(a(i+‘))&‘(a(i))~.
of A is defined to be the representation
Ad a o b = c (a)
a%S(a(*)),
a, b E A.
. . g&).
(2)
afforded by A itself
93
CASIMIR INVARIANTS FOR HOPF ALGEBRAS
Using equation (l), the following properties of Ad can be deduced Ad a o (bc) = x(Ad ab = &Ad
a(‘) o b)(Ad a(‘) o c),
(3)
a(l) o b)a(*)
(4)
(a) and since A is an algebra homomorphism
it follows that
Adao(Adboc)
= Ad(ab)oc.
The counit E : A --+ C determines a l-dimensional A-module V, we say that 21E V is an A-invariant if
(5)
representation
of A. Given any
Va E A.
av = c(a)w,
Thus we call c E A an invariant if it is an A-invariant under the adjoint action; i.e. Ada o c = E(U)C,
Vu E A.
We denote the space of all such invariants by C = {c E A : Ada o c = E(U)C, Va E A}. In view of equation (3), C is easily seen to be an algebra: In fact C is the centre of A; i.e. C = {c E A: ca = ac, V’a E A}. To see this, we note that if c is a central element, then Ad a o c = c
a%~@*))
= c
a(%(a(*))c
= E(O)C.
(a)
(a)
Conversely, if c E C then equation (4) implies UC= x(Ada(‘) (a)
o ~)a(*) = c @+a(*) (a)
= cu.
Thus we have shown LEMMA
1.
The central elements of A are precisely the A-invariants under the adjoint
action. In view of Schur’s lemma, the elements of C must reduce to scalar multiples of the identity on the finite-dimensional irreducible A-modules. If V is a finite-dimensional irreducible A-module with corresponding representation ‘IT,we denote the eigenvalue of c E C on V by x~(c). Then x* determines an algebra homomorphism
called an (infinitesimal) character. We may frequently use xrr, or equivalently, the eigenvalues of central elements, to provide a labelling of the finite-dimensional irreducible representations.
.I. R. LINKS and M. D. GOULD
94
Given any two A-modules V and W, let Z(V, W) denote the space of all linear mappings from V into W. Then it is easily seen that Z(V, W) becomes an A-module with the definition a 0 f = c a(‘)fS(a(2)), (a) i.e. (a 0 f)(V) = c a’1’f(s(a(2))v), (a) We thus call f E Z(V, W) A-invariant provided
f E qv, IV),
w E v.
(a Of)(u) = 4a)f(u). We have LEMMA
2. The A-invariants in Z(V, W) are precisely the A-module homomorphisms.
Proof: If f is an A-module homomorphism
(u 0 f)(V) = c &(s(aqV) (a)
then
= c a’l’s(aqf(V) (a)
= e(a)f(v),
i.e., f is an A-invariant. Conversely, if f is an A-invariant then f(aw) = c = ;
f(@))a%)
= c
E(a(l))f(a%)
&J o f(a’2’4
= ;
~U~~(~(~~2))~~3~~)
= &‘if(e(+) (a)
= r;(U). 0
In the case that W = C is the l-dimensional A-module with the definition (u 0 f)(U) = c
module, then V* = Z(V, C) becomes an
U”‘f(S(U’2’)W)
= ; E(&‘)f(s(u(~~)w) (a)
= f(S(u)w).
If 7r is the representation afforded by V, then the representation the dual of rr, is given by
rr* afforded by V*, called
K*(a) = 7?(S(a)), where t denotes matrix transposition. phism
With this definition, we have an A-module isomor-
5: v*@w+z(v,w),
CASIMIR INVARIANTS FOR HOPF ALGEBRAS
95
where We note that
f
I(f 8 w)(v) = (f, 4’wY Vf E V* is an A-invariant provided
cV*, WEW, VEV:
4a)f (v) = (a Of)(u) = f (S(a)v)
or equivalently, since c(a) = c(S(a)), e(a)f(v) = f(au).
For two A-modules V and W, a bilinear form t on V and W is equivalent to an element fc E (V @IW)’ defined by fc(w 63w) = [(v, w),
vu E v,
w E w.
Then ft is an A-invariant provided c(a)f[(v ~3w) = f[(A(a)w @w) = c f&z% (a)
~9ac2)w),
i.e. c(a)E(TJ,w) = c @%, (0) In particular, if a bilinear form ( , ) satisfies
Pw).
(V S(a)w) = (au, w)
(6)
then ( , ) is A-invariant .since C( a%, u(2)w) = C(V, S(u(‘))u(2)w) = E(U)(V,w). (a) (a) Consequently, we call a bilinear form satisfying equation (6) invariant. S-‘(u) we see that ( , ) is invariant if and only if (V, uw) = (S-‘(u)v,
Replacing a by
w).
In particular, a bilinear form on A itself is invariant provided (b, Ad S(u) o c) = (Ad a o b, c) or equivalently, (Ad S-‘(u) o b, c) = (6, Ad a o c). As we will see, it is the existence of such an invariant bilinear form on A which allows us to construct invariants. Finally, we say that a bilinear form on A is symmetric if
(a, b) = (b, S2(u)),
a, b E A.
Equivalently, the symmetry condition can be written as (a, b) = (s-2@), a) = (P(u),
S2(b)) = (s-2(u), s-2(b)).
J. R. LINKS and M. D. GOULD
96 3. Casimir
invariants
In this section we will demonstrate how to construct Casimir invariants for Hopf algebras which admit invariant bilinear forms. Throughout, by a submodule of a Hopf algebra A, we mean a subspace of A that is stable under the adjoint action. Let ( , ) be a fixed invariant bilinear form on A and let M C A be a non-trivial finite-dimensional submodule on which ( , ) is non-degenerate, i.e. for a E M. (a, M) = 0 or (M,a) = 0 + a = 0, Let {a,}zi be a basis for M and let {ui}zl be the corresponding form ( , ), i.e. (Ui) Uj) = sij.
dual basis under the
Note that we are defining {ui}gl to be a left dual basis. In the case that ( , ) is also symmetric, then we have s; = (a$ Uj) = (Uj, S2(u”)) so that {S2(ui)}g1 is the right dual basis. We define the Casimir element associated with m M by CM = C ui(ui). We have i=l
PROPOSITION
1. CM is in the centre
ofA.
Proof: It suffices to show in view of Lemma 1, that Ad x 0 CM = E(x)CM,
Mx E A.
Now for b E M, we can write b in terms of the basis elements b = e(b,ui)u”
= -j&‘,b)ui.
(7)
i=l
i=l
For x E A we have Ad x o CM = x(Ad x(l) o ui)(Ad xc2) o ai), (x1 where there is summation over repeated indices. Since M is stable under the adjoint action, equation (7) enables us to write the above as Ad x
o
CM = C(U”, Ad x(l) 0 a,)(Ad xc2) 0 U2, Uk)UjU"
=
;(“‘, Adxll'
o ui)(ui,AdS(x(2))
= ~(u~;Adx~l~oAdS(x(z~)ouk)u~ok = &$, (x1
Ad(x(‘)S(x(2))) o uk)uju”
= E(X)(Uj, U,+jU” = E(X)Cncr as required. 0
0 ak)ujak
CASIMIR INVARIANTS FOR HOPF ALGEBRAS
97
Remarks: In the case where A is the universal enveloping algebra of a simple Lie algebra L and M = L is the relevant Ad-module, then the above construction yields the universal Casimir element. Thus the invariants CM are generalizations of these. In the case of the group algebra of a finite group G and M is the linear span of a conjugacy class of G, we obtain invariants generating the centre (c.f. Gould [ll]).
We now turn our attention to Hopf algebras of trace-type and show that they admit invariant bilinear forms. We say that a Hopf algebra A is of trace-type if its antipode S is inner automorphic to its inverse; i.e. there exists u E A such that S(a) = uS-l(a)u-l or equivalently, S*(a) = uau-‘,
Va E A.
Suppose that r is a finite-dimensional representation We then have an invariant trace form defined by
of A with representation
space V.
(a, b) = tr, a(ab) = trr(uab), where we call tr, the u-trace. To see that this bilinear form is invariant we have (Ad a o b, c) = tr w(ua(‘)bS(a(*))c) = tr ~(S*(a(‘))ubS(a(*))c) = tr ~(zM(a(*))cS*(a(‘)))
= (b, Ad S(a) o c).
Thus if A is a Hopf algebra of trace-type we have an invariant bilinear form corresponding to any finite-dimensional representation r. Let ( , ) be such a form and let M E A be a finite-dimensional submodule (assumed irreducible w.1.o.g.) on which ( , ) is non-degenerate. Then we can construct the Casimir element GM. Under the assumption that M is irreducible, any invariant form on A reduces to a scalar multiple of the form ( , ) (Schur’s lemma) when restricted to M. Thus C M is unique up to scalar multiples, although it will depend upon the irreducible module M chosen. It is worth noting that u-traces satisfy the following properties: tr, 7r(S-‘(a))
= tr, 7r(S(a)),
tr, ?r(Ad a o b) = ~(a) tr, n(b), (a, b) = (b, S*(a));
@a) (gb) (gc)
i.e. ( , ) is necessarily symmetric. This in fact motivates our definition of a symmetric form. As well as admitting “second-order” Casimir invariants, Hopf algebras of trace-type provide us with a means of generating higher order invariants. Let r. be a hxed (but arbitrary) finite-dimensional irreducible representation of A with representation space Vo. We define an algebra homomorphism d : A -+ A 8 End VOby a(a) = (I 8 ro)A(a) = c a(‘) 8 ~~(a(*)), (a) and a mapping TV: A@EndVa--+A
(9)
J. R. LINKS and M. D. GOULD
98 bY
~%(a ~3 f)
(10)
= b(f)a
which we extend linearly. Note that we are defining trU(f) =
tr(no(u)f).
Now suppose that w = C ai ~3fi E A 8 End V, belongs to the centralizer of d(A); i.e. i
a(x)w = wd(x),
t/x E
(11)
A.
Then equation (11) is equivalent to
(12)
Ads o w = e(x)w, where we define Adz o w = c @(‘))w~9(S(x(*))) (I)
= c x(‘)a&~(~)) (~13
@ Ad x(*) o fi
and Adxof
= c so(x”‘)f~o(s(x(*~)). (I)
We have the following result generalizing Theorem ref. [9] (also c.f. refs. [5, lo]).
1 of ref. [16] and Proposition
1 of
PROPOSITION 2. Given w E A @ End V. sutisfLingequation (12), where A is a Hopf algebra of trace-type,ru(w) is in the cenm of A.
Proofi It suffices to show, in view of Lemma 1, that t/x E A.
Adz o r%(w) = e(x)rJw), To see this, we have emu
= T,(c(x)w) = r,(Adx =
o w)
c x(‘)c&(x(~)) tr,(Adx(*) (x),i
o fi)
= c ~(‘)a~S(x(~))e(x(*)) trU(fi) (x)4
from (8b)
=Xx (‘)a{ trU(fi)S(xc2)) (I)+ 0 = Adz o ru(w). Proposition 2 enables us to generate a family of invariants for A. If w belongs to the centralizer of a(A), then so do all of its higher powers so we obtain invariants CCrn) = TU(Wrn),
mEZ+.
99
CASIMIR INVARIANTS FOR HOPF ALGEBRAS
In particular, if A admits a Casimir element C M then we can use powers of I construct higher order invariants. An alternative method is to use powers of
to
where Q =
2
S-‘(ai) @ai
i=l
which can be shown to commute with the co-product action of A. 4. Quasi-triangular Hopf algebras In this section, we will briefly recall the definition of a quasi-triangular Hopf algebra and see that they are Hopf algebras of trace-type. For a Hopf algebra A, a twisting operation T : A @ A -+ A 18 A is introduced by T(a @b) = b @ a,
a,bc
A
(13)
which is extended linearly in the obvious way. An algebra homomorphism is then defined by
x : A + A @A
A(a) = T(A(a)) = c a(*) @ a(‘). (14) (a) Under this opposite co-product, A can be considered as a Hopf algebra with antipode 3 given by S(a) = s-i(a). (15) A is said to be quasi-triangulur if there exists an invertible element R E A @ A such that RA(a) = &)R, tla E A (16) and R satisfies (A @ I)R = Rt3R23, (I @ A)R = RnRn. In the above we have adopted the usual convention where, if R = C ai @ /Ii, then R~~=CN~@P~@I, i
R13 = C
CYI8 I@ pi
etc.
i
The importance of quasi-triangular Hopf algebras is that the element R, known as the universal R-matrix, satisfies the Yang-Baxter equation [4] RizRi3Rzs = RzsRi3R12. Suppose that A is quasi-triangular with R = C ai ~3 pi and R-l u = c S(/3i),i, we have the following results dui to Drinfeld [5]:
Up1= C S-‘(Si)CT’ zi
= C (Ti @ 6i. Setting ’
(17)
100
J. R. LINKS and M. D. GOULD
P(a) = uau-1,
Vu E A,
(18)
A(u) = u @3u(R%)-1,
(19)
where RT = T(R). For completeness, a proof of equation (19), different to the one given in [5], is presented in Appendix A. In view of equation (18) we immediately see that every quasi-triangular Hopf algebra is of trace-type. We should point out that the converse is not necessarily true; i.e. a Hopf algebra of trace-type is not necessarily quasi-triangular. A specific counter example is the dual algebra of the group algebra of a non-abelian group. Such an algebra is of trace-type but not quasi-triangular. For quasi-triangular Hopf algebras, the element RTR is of particular interest in the construction of invariants since it satisfies RTRA(a)
Vu E A.
= A(a)RTR,
It may be possible that RTR = I @ I, in which case A is said to be triangular. Important examples of quasi-triangular Hopf algebras arise from Drinfeld’s quantum double construction [4]. Such algebras are not generally triangular so that we may employ Proposition 2 to construct families of Casimir invariants from RTR = u 8 uA(&)
and its powers. We should mention that the element u satisfying equation (18) is only unique modulo the space of invertible central elements; i.e. if u satisfies equation (18) then so does uz where z is an invertible central element. From the point of view of taking u-traces, it may be more convenient to use such an element as seen in the next section on quantum groups. 5. Quantum groups In this section we wish to use the afore-mentioned theory in order to construct characteristic polynomial identities and generalized Gel’fand invariants for quantum groups. For every simple Lie algebra g, with Cartan matrix A = (aij), one can associate a quantum group U,(g). Let {(~i}T,r be the set of simple roots of g and let ( ,) be a fixed invariant bilinear form on H’, the dual of the Cartan subalgebra H of g, For a non-zero parameter q E C define qi = q’/*+%%),
/“‘l
= q*l/2hi,
hi E H.
Then U,(g) is generated by {Ic”, ei, fi}i=i with the following relations: pi,
kjl = 0,
[ei, fj]
lz(_l)t[1 -a1’] t=o
=
&w-teje; 92
bij
k?- kT2
qi -qil =
0,
’
i
f
j,
101
CASIMIR INVARIANT-S FOR HOPF ALGEBRAS
where (qrn - q-y(qm-1
- ql-“) . . . (q-+1 - q-m-l), (qn - q-n)(qn-1 - ql-“). . . (q - q-1)
1, 0,
m > n > o n=O
otherwise.
Mathematically, U,(g) has the structure of a Hopf algebra with co-product, and antipode defined, respectively, as
co-unit
A(!$‘) = ,F1 @ lCfr, A(e) = iii @ a + a @ I+,
E(kf1)= 1, S(k”)
= Icy,
a = ei,
fi,
(20)
c(fi) = 0,
6(ei) =
S(a) = -q-%qhp,
for a =
(21) ei, fi,
(22)
where p is the half sum of positive roots of g and h, is determined by p(hp) = (,u,p), Vp E H*. These mappings are extended uniquely to all elements of U,(g) such that A and E constitute algebra homomorphisms and S is an anti-automorphism. U,(g) also carries the structure of a Hopf algebra with the opposite co-product z and antipode 3 (cf. equations (14) and (15)). This structure is equivalent to replacing q by q-’ in equations (20) and (22). The algebra U,(g) contains two Hopf subalgebras U+!(g) and U;(g) generated by {ei, Ic”}~zl and {fi, k”}~zl, respectively. Let {e,} be a basis of U;(g) and {es} be the corresponding dual basis for U,‘(g). By the quantum double construction [4], U,(g) is quasi-triangular with R E U,(g) ~3 U,(g) given by
with inverse R-’
= (I ~3 S-l)R
= C e, ~3 S-‘(eS).
Then u = C S(es)e, and u-l =‘C S-2(es)e, s s P(u)
= ZLuu-1,
satisfy Va E Uq(g).
(23)
On the other hand, equation (22) implies S2(a) = q-2hPaq2hP,
Va E U,(g)
(24)
so that [5] c=u
-1q-2hp
E centre of U,(g).
When acting on a finite-dimensional irreducible representation takes the eigenvalue [5] x4(C) = q(“J+2p).
of highest weight .4, C (25)
102
J. R. LINKS and M. D. GOULD
Using this invariant, we can obtain characteristic polynomial identities and generalized Gel’fand invariants for U,(g). Throughout we let ~4, be a tied but arbitrary finite-dimensional irreducible representation. Let us define a matrix A E U,(g) 6x1End V(Ao) by A = (q - q-l)-l(I
@ I - C-’
8 rn,(C-‘)8(C)),
(26)
where d is given by equation (9). A can be considered as a matrix with entries from U,(g). When A acts on a finite-dimensional irreducible module V(A), it satisfies the polynomial identity
fitA- w(A)0
=
o,
(27)
i=l
where ai
= (q _ q-t)-r(1
_ *(X”Xi+2”+2p)-(“o,no+2f))
and {Xi}{=t is the set of distinct weights in V(Ao). position of the tensor product space is V(A) @ V(Ao)
= 6
(28)
To see this, suppose that the decom-
miV(Ai),
i=l
where rni is the multiplicity of V(A,). XA,(A)
= (4 -
When A acts on V(A,),
q-‘)-1(1
-
it takes the eigenvalue
x~(C-‘>X~,(C-‘)XA,(C>).
Using equation (25) and the fact that each Ai is of the form Ai=A+xi,
with X, some weight of V(Ao)
we have
Qi(A) = Xx,(A) = (q _ q-1)-1(1 _ q(X*,X~+2”+2p)-(“o,Ao+2p)), Equation (27) then follows. Note that as q --f 1 we have ai
which is identities different Since
+ $(Ao, A0 + 2~) - $(Xi, Xi + 211 + 2~)
(29)
the familiar result for the Lie algebra polynomial identities [13]. Thus our are a generalization of these; we note that the matrix A and identities (27) are to the ones previously obtained in ref. [9]. A centralizes a(u,(g)) we obtain the family of Casimir invariants Cc”) = rq(Am),
where TV= r+, which is given by equation (10). Here we have chosen for convenience to take q-traces rather than u-traces (cf. equations (23) and (24)) although they both lead to equivalent sets of invariants.
103
CASIMIR INVARIANTS FOR HOPF ALGEBRAS
When acting on a finite-dimensional value
irreducible module V(A), C’(m) takes the eigen-
(30) where ni is the multiplicity of Xi in V(.4,) and dim, V(A) is the q-dimension of V(A); i.e. dim, V(A) = tr .rm(q-2hp). The q-dimensions have been determined explicitly [lo] and are given by
dim,v(n) = II
q(“+PP)
_ q-(“+P>4
q(p,o)
_
q-(p,“)
(31)
’
CrEQ
where 9 denotes the set of all positive roots of the underlying Lie algebra g. The proof of the eigenvalue formula for equation (30) is identical to that of Gould et al. [9] (see also ref. [12]). Unlike the invariants of reference [9], the invariants above reduce in the q + 1 limit to the invariants constructed by Gould [12] for simple Lie algebras which includes the Gel’fand invariants of the classical Lie algebras as a special case. The matrix A defined by equation (26) can in fact be determined explicitly. Note that equations (19) and (22) imply that A(C) = (C @ C)R%. Thus A is also given by A = (q - q-l)-‘(I
@ I - RzoR,,,),
where R$
Rn, = (I @ TA,)R,
= (I @ nn,)RT.
A formulation for R has been obtained by Kirillov and Reshetikhin [17] which reads R
=
,cb, hi@h’
where K = i(dimg Ln]
q
=
- T),
Q”- q-” q_q-’
’
qp =
qw*MPMP))
[nlq! = [nlq[n - llq.. . [llq, r
where a+)
is defined by [hi, Ep] = (ai, a(p))Ep
and hi satisfies 2 p(hi)v(hi)
= (p, v)
i=l
for all p, v E H’. The Ep and Fp are defined in ref. [17] and we refer the reader to that paper for details. The formula for R allows us to then determine A which is
104
J. R. LINKS
and M. D. GOULD
given by
@ rAo(E;n’ . . . E,mK) 1
We remark that since that representation 7rn0 is finite-dimensional, the sums over (2) and {m} truncate to finite sums. The matrix A defined by equation (26) is not the only matrix which gives us an analogue of the characteristic matrix. If we define the matrices A’ = (q - q-1)-1(1@ B = (4 -
B’ =
q-‘)-ye
(q - q-‘)(C
I -
C-l @~A,(c-l)s(c)) = (q - q-y(I@
c3 7rn,(c)a(c-‘) -
8 ~n,(C)a(c-‘)
I -
RnoR;o),
I @I) = (q - q-‘)-‘(R;;,‘(R-‘);o
- I CSJ I) = (q -
q-l)-l((R-l);oR$
- I @ I),
-
I CZJ I),
where 8 is defined by B(a) = (I @ 7Q,)&-L) = c cJ2) @ 7r~&(‘)), (a) then A’ satisfies the same identity (27) as A while B and B’ satisfy
n(B -
Pi(A)Q = 0,
i=l
where Pi(A) is given by pi(n) = (q _ q-l)-‘(q-(X”Xi+2”+2p)+(Ao,Ao+2P) _ I). Consequently, these matrices will also allow us to obtain sets of Casimir invariants. The matrix B lies in the centralizer of a(U,(g)) so we obtain a set of invariants by D+)
= TV.
When acting on V(n), these invariants take the eigenvalues (32) i=l
These eigenvalues are clearly different from those given by equation (30) so we must conclude that {C(“)} and {DC”)} are two different sets of invariants. However as q 4 1,
CASIMIR INVARIANTS FOR HOPF ALGEBRAS
105
xn(C(“)) and xn(D(“)) both re d uce to the same eigenvalue formula, so the two sets of invariants are equivalent in this limit. The matrices A’ and B’ both lie in the centralizer of a(U,(g)). Under the co-product a, U,(g) has a Hopf algebra structure with antipode 3 = S-l. Thus the square of the antipode is given by -2 s (a)
=
s-2(a)
=
2b (
q2h%q-
v’a E Udl?).
This allows us to construct the sets of invariants C’(“) = T&(A’)“),
D’(“)
= q_l((B’)y,
where T~.-~ E rg2,,,. It is not hard to deduce that the eigenvalues of C’(“) are given by equation (30) while those of D’(m) are given by equation (32), bearing in mind that the q-dimensions are invariant under q + q-’ (equation (31)). Thus we can conclude that C(“) = C’(4, D(“) = D’(m). Using these results we are now in a position to determine the images of these invariants under the action of the antipode S. Let A{ denote the highest weight of the dual representation of TA,; i.e. r/l;(e)
= 7r&(S(a)).
We let ‘;i:be the matrix obtained by replacing XA, with 7r,r(;in the definition of A, and we let C@) denote the corresponding R$RA,
=
C
et%
invariants. We then have @
7TAo(etes)
t,s = c
etes @I ?r~$S-‘(etes))
= (S o I)(RA;R~.)~.
t,s
It follows that A” = (S 18 I)((Ti’)m)t
A = (S @ I)(A’)t,
from which we obtain C(“) = T,(A”)
= (I@ tr)(l@
= (s ‘8 I)(1 8 tr)(l = S(T,&iy)
r&-2hp))(s
@II)((SI’)m)t
@ 7rAb; (q2hp))(A’)m
= SpQ).
Similarly, we also have the result for the invariants arising from the matrix B; i.e. DC”) = S(D(“)). To conclude this section, we briefly mention how projection operators can be constructed from the matrix A (or A’, B, B’). Acting on V(A), we have that A satisfies equation (27); i.e. h(A i=l
- cQ.4)1)
= 0.
106
J. R. LINKS and M .J. GOULD
Let {y,(A)}:=,
be the maximal set of distinct ai
and let P, be defined by
( A-y&v
p, = fj
).
%(A) - -d4
s=l
s#r It is easily verified that these operators satisfy the relations t
PA
c
= &tPt,
P,=I@I
T=I
and P,. projects onto the irreducible representations of the tensor product space with highest weight A + A,. These projection operators play an important role in obtaining solutions to the Yang-Baxter equation [4, 61. 6. U,(gl(n)) invariants
and polynomial
identities
In this section we will apply the general theory we have developed to U,(gl(n)). For this case we are able to obtain generalized Gel’fand invariants by taking q-traces of the matrix A = (q - q-‘)-‘(I
@ I - R; _&j-i))
and its powers. Here we have chosen the contragredient reference representation, which is defined by Q,-l)(a)
= &$Y(e)),
where y is the principal anti-automorphism generators of U,(gl(n)) by -/(JQ*1)
=
vector representation
(33) as our
a E U,(gl(n)), of U,(gl(n)). r(qkEii)
-J%,i*1,
y is defined on the elementary =
q~&i
and is related to the antipode by -/(a) = qhp s(a)q+. In order to determine
A explicitly we observe that
R =
c
e, ~3 es=
s
c
r(e,)
C3r(es).
.9
Thus
R= R(l.6)= (I @“(1,6))
c de,) @n(l,b)(y(es))= (Y 8 V$6,_1,. s
The matrix Rc1,6) has been obtained by Jimbo [7] and is given by
(34)
107
CASIMIR INVARIANTS FOR HOPF ALGEBRAS
where zji
(q - q-‘)q1/2(E~~+E3j-1)Ejji, qEjj,
=
i # j . . 2=j
and Eji is defined recursively as jSkSi.
Eii = EjkEki - q_lEkiEjk,
Here, eij are the standard generators of gl(n) in the fundamental vector representation. From equation (34) we then have
or
j>i
where we have used the fact that r*(u) = a, Va E U,(gl(n)). induction that T(&j)
1
=
q-(Pt-‘i-E3)~ij,
i
<
q(PyEi--Zj)kii,
i
1 j,'
It is easily verified by
j
(35)
-
where
Eii =
-_‘~ - q-‘)q-‘/*(E”“+Ejj-‘)E~i,
i z j
Q
2=j
. -7
.
and Eij is defined recursively as Eij = E,(,Ekj - qEhjElk,
isksj,
%*1
=
Ei,i*l*
In equation (35) we have used the standard basis for the weights of U,-,(gl(n)) satisfying (Ei,
Ej)
=
bij
and p is given by p
=-
1 n *C( n + 1 -2i)Ei. i=l
The matrix Rc6,_1) is then given by
R(lll-1)
=
f:
q(Pt’r4gi
@
eii.
(36)
j>i
By hermiticity we can deduce that R$ _l) is given by
Rg,-l)=cq-_(P7ej-Ei)Eji j
@
eii
(37)
108
J. R. LINKS and M. D. GOULD
and so substituting equations (36) and (37) into (33) we get that the entries of the matrix A are given by Aij = (4 _ 4-i)-i
(bij _ 2
q(P’2C~-i’-“)EieEkj)
k=iVj
=
c4 _
q-l)-l
(sij
_
2
+i+j-2k)&jjkj),
k=iVj
where a V b = max(a, b). It is easily checked that as q -+ 1, Aij -+ Eij, the generators of gl(n), so we have an analogue of the characteristic vector matrix of gl(n). Acting on an irreducible module V(A), A satisfies the polynomial identity fi(A
- cq(A)I)
= 0,
i=l
where &(A) = (q _ q-‘)-i(l
_ q-W”+“-i9
and Ai is given by LIP= (A, Q). This polynomial identity allows us to construct projection operators in the following way. First note that if A is a dominant weight then the w(A) are distinct; i.e. ai
= q(A)
=+ i = j.
Thus we can construct the operators P, =
satisfying n P,Pt = ws,
c r=l
P, = I@ I.
To determine the generalized Gel’fand invariants, we first note that (a~g,_l)(Q-2hp))ij = SijQ(n+1-2i)eii. The invariants are then given by Cc”) = rq(Am) =
C i,j,k . . . . 7
q(“+1-2i) AijAjk . . . A,i , \ m terms
which reduce to the usual gl(n) Gel’fand invariants as g -+ 1. Acting on an irreducible module V(A), these invariants take the eigenvalues
109
CASIMIR INVARIANTS FOR HOPF ALGEBRAS
Substituting equation (31) for the q-dimensions and using the fact that for gl(n), every a E @ is of the form CY= Ei - Cj,
i
we arrive at
XA(@))= (q -
q-1)-m
?(I
_
q-2(A;+n-i))m
fi
i=l
j=l j#i
q(At-Aj-l+j-i) q(Ai-A,+j-i)
_
q-(Ai-Aj-l+j-i)
_
q-(“i-“j+j-i)
.
We remark that in the limit q --f 1 the above formula reduces to the usual one for the eigenvalues of the gl(n) Gel’fand invariants as required. Finally, we note that the entries Aij of the matrix A span a submodule M of U,(gl(n)) under the adjoint action. In fact, the operators Aij transform as an adjoint tensor operator [i.e. as (1,6) 8 (0, -l)]. A n invariant bi-linear form which is non-degenerate on A4 may then be set up according to (-&j, AM) = tr~~,,0~(q-2hPAijA~~). Utilizing the construction of Proposition q2nC(2) = q2”
1 we arrive at the quadratic invariant
2
q(n+‘-2i)&Aji.
i,j=l
7. Conclusion In this paper we have developed some methods for constructing invariants for Hopf algebras. It was shown that there is a class of Hopf algebra (referred to as trace-type) which always admit invariant bilinear forms and provide a means of generating families of invariants. Due to a result of Drinfeld [5], it can be seen that all quasi-triangular Hopf algebras fall into this class. Using our results we were able to determine generalized Gel’fand invariants and characteristic polynomial identities for quantum groups. Their eigenvalues and characteristic roots have been computed and they are expressed in terms of the highest weights of the irreducible representations. These invariants and identities are not the only possible ones (see [9]) but we feel that they will be the easiest to work with because of the simplicity of their construction and eigenvalue formulae. We have’ also determined the images of these invariants under the antipode. To illustrate our results we have studied V,(gI(n)) in detail. Appendix A
In this appendix we derive the result (cf. equation (19)) A(&)
= u-’ 8 u-lRTR
E D(A) @ID(A),
where D(A) is the quasi-triangular Hopf algebra arising from Drinfeld’s quantum double construction [4]. Drinfeld showed that any Hopf algebra A may be naturally embedded
110
J. R. LINKS and M. D. GOULD
in a quasi-triangular Hopf algebra D(A). Let {e,} be a basis for A with multiplication, co-product and antipode given by T eset = mater, S(e,) = &t, A(es) = ,$et 8 e,, (38) where summation is taken over repeated indices. Let A0 denote the algebra dual of A, with corresponding dual basis {es} satisfying eSet = &rer,
A(eS) = rnFrer@ et,
S(e’) = (v-‘)ie”.
(39)
A and A0 can be identified as subalgebras of D(A) G A @ A0 in a natural way. We identify a E A with Z = a @ I and b E A0 with 3 = I @ b. The elements Z,, ES obey the relations (38) and (39), respectively. Multiplication between the elements {e,} and {es} is defined by - -t = e,@e’, e,e es& = ei m&m~ip~pjnr(V-l)Hej
@
= m~,m~i~:“~jr’(v-‘)lQejei. Drinfeld [4] has shown that the element R = Z, 63Z? satisfies RA(a) = &(a)R, (A 8 I)R = RnR23,
a E D(A) (I ~!aA)R = &d&2
(40)
and is invertible with inverse given by R-’ = S(E,) @ES so that D(A) is quasi-triangular.
It follows that 21= S(Z)&
$(a) As a consequence D(A) [18]:
= 21au-i,
and u-i = S2(Z?)F,
a E D(A).
satisfy (41)
of equation (40), the following result can be deduced to hold in ~~qm&&q
= p~qm~ti$.
(42)
Using equations (2) and (41) we conclude that A(S2(u)) = 218 uA(a)z~-’ @ u-l. Equations (42) and (43) will enable us to show A@-‘) = 21-l @ u-‘RTR. Indeed we have, by direct calculation A(&)
= A(S-2(i?)Q =2L -’ @ K’A(Z)~ @ 2~A(??s) from (42) =U -’ 8 ~-lm&Zr @Ftu @ uA(Z,) =u -’ @ u-‘Z
C% etu @ ~A(&&.)
=U -’ @ u-‘Z
@Ztu @ u&‘el @e,A(e,)
(43)
CASIMIR
INVARIANTS
FOR HOPF
ALGEBRAS
111
REFERENCES
PI PI [31 [41 PI PI [71 PI 191 IlOl PII WI u31 u41 PI WI u71 P81
R. J. Baxter: Exactly Solved Models in Statistical Mechanics, Academic Press, London, 1982. P. P. Kulish and E. K. Sklyanin: in “Integrable Quantum Field Theories”, Lecrure Notes in Physics 151, 61, Springer-Verlag, New York, 1982. E. Witten: Commun. Math. Phys. 121 (1989), 351; M. Wadati, T. Deguchi and Y. Akutsu: Phys. Rep. 180 (1989) 248. V. G. Drinfeld: Proc. I.C.M., Berkeley 1 (1986), 798. V. G. Drinfeld: Algebra and Analysis (in Russian) 1, No. 2 (1989), 30. M. Jimbo: Lett. Mafh. Phys. 10 (1985) 63. M. Jimbo: Left. Math. Phys. 11 (1986) 247. M. Rosso: Commun. Math. Phys. 117 (1988), 581. M. D. Gould, R. B. Zhang and A. J. Bracken: Generalized Gellfand Invariants and Characteristic Identities for Quantum Groups, University of Queensland preprint (Nov. 1989). R. B. Zhang, M. D, Gould and A. J. Bracken: Quantum Group Invariants and Link Polynomials, University Queensland preprint (Nov. 1989). M. D. Gould: J. Phys. A: Math. Gen. 20 (1987), 2657. M. D. Gould: Rep. Math. Phys. 27 (1989), 73. A. J. Bracken and H. S. Green: J. Math. Phys. 12 (1971), 2099; H. S. Green: J. Math. Phys. 12 (1971) 2106. M. E. Sweedler: Hopf Algebras, Benjamin, New York, 1969. E. Abe: Hopf Algebras, Cambridge University Press, 1980. L. D. Faddeev, N. Yu. Reshetikhin and L. A. Takhtajan: Algebraic Analysis 1 (1) (1989) 129. A. N. Kirillov and N. Yu. Reshetikhin: Commun. Math. Phys. 134 (1990), 421. N. Yu. Reshetikhin: Quantized Universal Enveloping Algebras of the Yang-Baxter Equation and Invariants of Links I, II, L.O.M.I. (Leningrad) preprint E-4-87, E-17-87.
REPORTS
ON MATHEMATICAL
CASIMIR INVARIANTS
PHYSICS
No. 1
FOR HOPF ALGEBRAS
J. R. LINKS and M. D. GOULD Department of Mathematics, University of Queensland, St. Lucia, QLD, Australia 4067 (Received May 14, 1991) A full set of invariants for an arbitrary quantum group is constructed which reduce to the Gel’fand invariants of the corresponding Lie algebra in the zero deformation limit q -+ 1. The eigenvalues of these invariants on irreducible representations are also determined as a simple function of the highest weight labels. Extensions to more general classes of Hopf algebra are also discussed.
1. Introduction
Recently, there has been renewed interest in the study of Hopf algebras because of their role in obtaining solutions to the Yang-Baxter equation, which arises in areas such as integrable lattice models [l], the quantum inverse scattering method [2] and the theory of knots and links [3]. In particular, it is the so called quasi-triangular Hopf algebras as defined by Drinfeld [4] that have received most attention since they admit a canonical element known as the universal (i.e. representation independent) R-matrix which automatically provides a solution to the Yang-Baxter equation. From the point of view of physical applications of quasi-triangular Hopf algebras, it is necessary to study their representation theory. This in turn leads us to a study of the centralizer of these algebras, as we expect that knowledge of the centralizer will prove to be useful in their general representation theory. In this paper we investigate some methods of constructing invariants for Hopf algebras. Quantum groups [4, 61 are examples of quasi-triangular Hopf algebras which are obtained by deforming the universal enveloping algebra of a simple Lie algebra through the introduction of a non-zero deformation parameter q. As q -+ 1, the defining relations of a quantum group reduce to those of a simple Lie algebra. The representation theory of quantum groups appears to be very similar to that of simple Lie algebras. In particular, it is now known [8] that when q is not a root of unity, all finite dimensional representations of quantum groups are completely reducible and the irreducible ones can be classified in terms of their highest weights. One of the main aims of this paper is to determine the analogues of the Gel’fand invariants for quantum groups. A set of invariants generating the centre of the quantum group, which do not give rise to the Gel’fand invariants of the corresponding Lie algebra as q + 1, has already been determined in general terms by Gould et al. [9] and Faddeev et al. [16]. In this paper we construct an alternative set of invariants (and their eigenvalues) which reduce [911
J. R. LINKS and M. D. GOULD
92
to the Gel’fand invariants as 4 -+ 1. We follow the general approach of ref. [9] although our construction is simpler and more direct; in particular, fully explicit results are obtained for the generalized Gel’fand invariants of U,(gl(n)). We expect that these invariants will be easier to use for both practical and theoretical considerations. In addition, we will determine the images of these generalized Gel’fand invariants under the action of the antipode, and determine characteristic polynomial identities for quantum groups which are generalizations of those obtained by Bracken and Green [13] for simple Lie algebras. These polynomial identities allow us to construct projection operators which play an important role in constructing solutions of the Yang-Baxter equation [4, 61. This paper begins with some general results on Hopf algebras and invariant bilinear forms. We will show that there is a particular class of Hopf algebras which always admit invariant bilinear forms. Such a form affords a unified construction of Casimir invariants, as we will see. All quasi-triangular Hopf algebras fall into this class so the theory is readily applicable to them. In this case however, the universal R-matrix enables an alternative construction of Casimir invariants. This general theory is then used to determine the generalized Gel’fand invariants and characteristic polynomial identities for quantum groups. As an example, we will consider U,(gl(n)) in detail. 2. Fundamentals Let A denote a Hopf algebra with co-product A : A --f A @ A, bijective antipode S : A -+ A and co-unit E : A + C, where C is the underlying complex field. We will not give the full defining relations of a Hopf algebra here but refer the reader to refs. [14, 151 for details. However, using the Sigma notation of Sweedler [14] we note the following important properties:
c
CL(l) @. . . @ ,(,G)),@+l)
@ . . . @ @+l)
(a)
=
c
u(l) @. . . 63 u(n)
=g
u(l) @. .
=g
a(l) @ . . . @ &)q&+‘$-&i+*)
~3 A(&))
@ . . . @ c&n-1) @ . . . gJ a(n+*),
(1)
(a)
c a(‘)@
. . . @A(S(di’))@.
. @~a(~-~) = c a(%. (a)
(a)
The adjoint representation under the action
. . cm(a(i+‘))&‘(a(i))~.
of A is defined to be the representation
Ad a o b = c (a)
a%S(a(*)),
a, b E A.
. . g&).
(2)
afforded by A itself
93
CASIMIR INVARIANTS FOR HOPF ALGEBRAS
Using equation (l), the following properties of Ad can be deduced Ad a o (bc) = x(Ad ab = &Ad
a(‘) o b)(Ad a(‘) o c),
(3)
a(l) o b)a(*)
(4)
(a) and since A is an algebra homomorphism
it follows that
Adao(Adboc)
= Ad(ab)oc.
The counit E : A --+ C determines a l-dimensional A-module V, we say that 21E V is an A-invariant if
(5)
representation
of A. Given any
Va E A.
av = c(a)w,
Thus we call c E A an invariant if it is an A-invariant under the adjoint action; i.e. Ada o c = E(U)C,
Vu E A.
We denote the space of all such invariants by C = {c E A : Ada o c = E(U)C, Va E A}. In view of equation (3), C is easily seen to be an algebra: In fact C is the centre of A; i.e. C = {c E A: ca = ac, V’a E A}. To see this, we note that if c is a central element, then Ad a o c = c
a%~@*))
= c
a(%(a(*))c
= E(O)C.
(a)
(a)
Conversely, if c E C then equation (4) implies UC= x(Ada(‘) (a)
o ~)a(*) = c @+a(*) (a)
= cu.
Thus we have shown LEMMA
1.
The central elements of A are precisely the A-invariants under the adjoint
action. In view of Schur’s lemma, the elements of C must reduce to scalar multiples of the identity on the finite-dimensional irreducible A-modules. If V is a finite-dimensional irreducible A-module with corresponding representation ‘IT,we denote the eigenvalue of c E C on V by x~(c). Then x* determines an algebra homomorphism
called an (infinitesimal) character. We may frequently use xrr, or equivalently, the eigenvalues of central elements, to provide a labelling of the finite-dimensional irreducible representations.
.I. R. LINKS and M. D. GOULD
94
Given any two A-modules V and W, let Z(V, W) denote the space of all linear mappings from V into W. Then it is easily seen that Z(V, W) becomes an A-module with the definition a 0 f = c a(‘)fS(a(2)), (a) i.e. (a 0 f)(V) = c a’1’f(s(a(2))v), (a) We thus call f E Z(V, W) A-invariant provided
f E qv, IV),
w E v.
(a Of)(u) = 4a)f(u). We have LEMMA
2. The A-invariants in Z(V, W) are precisely the A-module homomorphisms.
Proof: If f is an A-module homomorphism
(u 0 f)(V) = c &(s(aqV) (a)
then
= c a’l’s(aqf(V) (a)
= e(a)f(v),
i.e., f is an A-invariant. Conversely, if f is an A-invariant then f(aw) = c = ;
f(@))a%)
= c
E(a(l))f(a%)
&J o f(a’2’4
= ;
~U~~(~(~~2))~~3~~)
= &‘if(e(+) (a)
= r;(U). 0
In the case that W = C is the l-dimensional A-module with the definition (u 0 f)(U) = c
module, then V* = Z(V, C) becomes an
U”‘f(S(U’2’)W)
= ; E(&‘)f(s(u(~~)w) (a)
= f(S(u)w).
If 7r is the representation afforded by V, then the representation the dual of rr, is given by
rr* afforded by V*, called
K*(a) = 7?(S(a)), where t denotes matrix transposition. phism
With this definition, we have an A-module isomor-
5: v*@w+z(v,w),
CASIMIR INVARIANTS FOR HOPF ALGEBRAS
95
where We note that
f
I(f 8 w)(v) = (f, 4’wY Vf E V* is an A-invariant provided
cV*, WEW, VEV:
4a)f (v) = (a Of)(u) = f (S(a)v)
or equivalently, since c(a) = c(S(a)), e(a)f(v) = f(au).
For two A-modules V and W, a bilinear form t on V and W is equivalent to an element fc E (V @IW)’ defined by fc(w 63w) = [(v, w),
vu E v,
w E w.
Then ft is an A-invariant provided c(a)f[(v ~3w) = f[(A(a)w @w) = c f&z% (a)
~9ac2)w),
i.e. c(a)E(TJ,w) = c @%, (0) In particular, if a bilinear form ( , ) satisfies
Pw).
(V S(a)w) = (au, w)
(6)
then ( , ) is A-invariant .since C( a%, u(2)w) = C(V, S(u(‘))u(2)w) = E(U)(V,w). (a) (a) Consequently, we call a bilinear form satisfying equation (6) invariant. S-‘(u) we see that ( , ) is invariant if and only if (V, uw) = (S-‘(u)v,
Replacing a by
w).
In particular, a bilinear form on A itself is invariant provided (b, Ad S(u) o c) = (Ad a o b, c) or equivalently, (Ad S-‘(u) o b, c) = (6, Ad a o c). As we will see, it is the existence of such an invariant bilinear form on A which allows us to construct invariants. Finally, we say that a bilinear form on A is symmetric if
(a, b) = (b, S2(u)),
a, b E A.
Equivalently, the symmetry condition can be written as (a, b) = (s-2@), a) = (P(u),
S2(b)) = (s-2(u), s-2(b)).
J. R. LINKS and M. D. GOULD
96 3. Casimir
invariants
In this section we will demonstrate how to construct Casimir invariants for Hopf algebras which admit invariant bilinear forms. Throughout, by a submodule of a Hopf algebra A, we mean a subspace of A that is stable under the adjoint action. Let ( , ) be a fixed invariant bilinear form on A and let M C A be a non-trivial finite-dimensional submodule on which ( , ) is non-degenerate, i.e. for a E M. (a, M) = 0 or (M,a) = 0 + a = 0, Let {a,}zi be a basis for M and let {ui}zl be the corresponding form ( , ), i.e. (Ui) Uj) = sij.
dual basis under the
Note that we are defining {ui}gl to be a left dual basis. In the case that ( , ) is also symmetric, then we have s; = (a$ Uj) = (Uj, S2(u”)) so that {S2(ui)}g1 is the right dual basis. We define the Casimir element associated with m M by CM = C ui(ui). We have i=l
PROPOSITION
1. CM is in the centre
ofA.
Proof: It suffices to show in view of Lemma 1, that Ad x 0 CM = E(x)CM,
Mx E A.
Now for b E M, we can write b in terms of the basis elements b = e(b,ui)u”
= -j&‘,b)ui.
(7)
i=l
i=l
For x E A we have Ad x o CM = x(Ad x(l) o ui)(Ad xc2) o ai), (x1 where there is summation over repeated indices. Since M is stable under the adjoint action, equation (7) enables us to write the above as Ad x
o
CM = C(U”, Ad x(l) 0 a,)(Ad xc2) 0 U2, Uk)UjU"
=
;(“‘, Adxll'
o ui)(ui,AdS(x(2))
= ~(u~;Adx~l~oAdS(x(z~)ouk)u~ok = &$, (x1
Ad(x(‘)S(x(2))) o uk)uju”
= E(X)(Uj, U,+jU” = E(X)Cncr as required. 0
0 ak)ujak
CASIMIR INVARIANTS FOR HOPF ALGEBRAS
97
Remarks: In the case where A is the universal enveloping algebra of a simple Lie algebra L and M = L is the relevant Ad-module, then the above construction yields the universal Casimir element. Thus the invariants CM are generalizations of these. In the case of the group algebra of a finite group G and M is the linear span of a conjugacy class of G, we obtain invariants generating the centre (c.f. Gould [ll]).
We now turn our attention to Hopf algebras of trace-type and show that they admit invariant bilinear forms. We say that a Hopf algebra A is of trace-type if its antipode S is inner automorphic to its inverse; i.e. there exists u E A such that S(a) = uS-l(a)u-l or equivalently, S*(a) = uau-‘,
Va E A.
Suppose that r is a finite-dimensional representation We then have an invariant trace form defined by
of A with representation
space V.
(a, b) = tr, a(ab) = trr(uab), where we call tr, the u-trace. To see that this bilinear form is invariant we have (Ad a o b, c) = tr w(ua(‘)bS(a(*))c) = tr ~(S*(a(‘))ubS(a(*))c) = tr ~(zM(a(*))cS*(a(‘)))
= (b, Ad S(a) o c).
Thus if A is a Hopf algebra of trace-type we have an invariant bilinear form corresponding to any finite-dimensional representation r. Let ( , ) be such a form and let M E A be a finite-dimensional submodule (assumed irreducible w.1.o.g.) on which ( , ) is non-degenerate. Then we can construct the Casimir element GM. Under the assumption that M is irreducible, any invariant form on A reduces to a scalar multiple of the form ( , ) (Schur’s lemma) when restricted to M. Thus C M is unique up to scalar multiples, although it will depend upon the irreducible module M chosen. It is worth noting that u-traces satisfy the following properties: tr, 7r(S-‘(a))
= tr, 7r(S(a)),
tr, ?r(Ad a o b) = ~(a) tr, n(b), (a, b) = (b, S*(a));
@a) (gb) (gc)
i.e. ( , ) is necessarily symmetric. This in fact motivates our definition of a symmetric form. As well as admitting “second-order” Casimir invariants, Hopf algebras of trace-type provide us with a means of generating higher order invariants. Let r. be a hxed (but arbitrary) finite-dimensional irreducible representation of A with representation space Vo. We define an algebra homomorphism d : A -+ A 8 End VOby a(a) = (I 8 ro)A(a) = c a(‘) 8 ~~(a(*)), (a) and a mapping TV: A@EndVa--+A
(9)
J. R. LINKS and M. D. GOULD
98 bY
~%(a ~3 f)
(10)
= b(f)a
which we extend linearly. Note that we are defining trU(f) =
tr(no(u)f).
Now suppose that w = C ai ~3fi E A 8 End V, belongs to the centralizer of d(A); i.e. i
a(x)w = wd(x),
t/x E
(11)
A.
Then equation (11) is equivalent to
(12)
Ads o w = e(x)w, where we define Adz o w = c @(‘))w~9(S(x(*))) (I)
= c x(‘)a&~(~)) (~13
@ Ad x(*) o fi
and Adxof
= c so(x”‘)f~o(s(x(*~)). (I)
We have the following result generalizing Theorem ref. [9] (also c.f. refs. [5, lo]).
1 of ref. [16] and Proposition
1 of
PROPOSITION 2. Given w E A @ End V. sutisfLingequation (12), where A is a Hopf algebra of trace-type,ru(w) is in the cenm of A.
Proofi It suffices to show, in view of Lemma 1, that t/x E A.
Adz o r%(w) = e(x)rJw), To see this, we have emu
= T,(c(x)w) = r,(Adx =
o w)
c x(‘)c&(x(~)) tr,(Adx(*) (x),i
o fi)
= c ~(‘)a~S(x(~))e(x(*)) trU(fi) (x)4
from (8b)
=Xx (‘)a{ trU(fi)S(xc2)) (I)+ 0 = Adz o ru(w). Proposition 2 enables us to generate a family of invariants for A. If w belongs to the centralizer of a(A), then so do all of its higher powers so we obtain invariants CCrn) = TU(Wrn),
mEZ+.
99
CASIMIR INVARIANTS FOR HOPF ALGEBRAS
In particular, if A admits a Casimir element C M then we can use powers of I construct higher order invariants. An alternative method is to use powers of
to
where Q =
2
S-‘(ai) @ai
i=l
which can be shown to commute with the co-product action of A. 4. Quasi-triangular Hopf algebras In this section, we will briefly recall the definition of a quasi-triangular Hopf algebra and see that they are Hopf algebras of trace-type. For a Hopf algebra A, a twisting operation T : A @ A -+ A 18 A is introduced by T(a @b) = b @ a,
a,bc
A
(13)
which is extended linearly in the obvious way. An algebra homomorphism is then defined by
x : A + A @A
A(a) = T(A(a)) = c a(*) @ a(‘). (14) (a) Under this opposite co-product, A can be considered as a Hopf algebra with antipode 3 given by S(a) = s-i(a). (15) A is said to be quasi-triangulur if there exists an invertible element R E A @ A such that RA(a) = &)R, tla E A (16) and R satisfies (A @ I)R = Rt3R23, (I @ A)R = RnRn. In the above we have adopted the usual convention where, if R = C ai @ /Ii, then R~~=CN~@P~@I, i
R13 = C
CYI8 I@ pi
etc.
i
The importance of quasi-triangular Hopf algebras is that the element R, known as the universal R-matrix, satisfies the Yang-Baxter equation [4] RizRi3Rzs = RzsRi3R12. Suppose that A is quasi-triangular with R = C ai ~3 pi and R-l u = c S(/3i),i, we have the following results dui to Drinfeld [5]:
Up1= C S-‘(Si)CT’ zi
= C (Ti @ 6i. Setting ’
(17)
100
J. R. LINKS and M. D. GOULD
P(a) = uau-1,
Vu E A,
(18)
A(u) = u @3u(R%)-1,
(19)
where RT = T(R). For completeness, a proof of equation (19), different to the one given in [5], is presented in Appendix A. In view of equation (18) we immediately see that every quasi-triangular Hopf algebra is of trace-type. We should point out that the converse is not necessarily true; i.e. a Hopf algebra of trace-type is not necessarily quasi-triangular. A specific counter example is the dual algebra of the group algebra of a non-abelian group. Such an algebra is of trace-type but not quasi-triangular. For quasi-triangular Hopf algebras, the element RTR is of particular interest in the construction of invariants since it satisfies RTRA(a)
Vu E A.
= A(a)RTR,
It may be possible that RTR = I @ I, in which case A is said to be triangular. Important examples of quasi-triangular Hopf algebras arise from Drinfeld’s quantum double construction [4]. Such algebras are not generally triangular so that we may employ Proposition 2 to construct families of Casimir invariants from RTR = u 8 uA(&)
and its powers. We should mention that the element u satisfying equation (18) is only unique modulo the space of invertible central elements; i.e. if u satisfies equation (18) then so does uz where z is an invertible central element. From the point of view of taking u-traces, it may be more convenient to use such an element as seen in the next section on quantum groups. 5. Quantum groups In this section we wish to use the afore-mentioned theory in order to construct characteristic polynomial identities and generalized Gel’fand invariants for quantum groups. For every simple Lie algebra g, with Cartan matrix A = (aij), one can associate a quantum group U,(g). Let {(~i}T,r be the set of simple roots of g and let ( ,) be a fixed invariant bilinear form on H’, the dual of the Cartan subalgebra H of g, For a non-zero parameter q E C define qi = q’/*+%%),
/“‘l
= q*l/2hi,
hi E H.
Then U,(g) is generated by {Ic”, ei, fi}i=i with the following relations: pi,
kjl = 0,
[ei, fj]
lz(_l)t[1 -a1’] t=o
=
&w-teje; 92
bij
k?- kT2
qi -qil =
0,
’
i
f
j,
101
CASIMIR INVARIANT-S FOR HOPF ALGEBRAS
where (qrn - q-y(qm-1
- ql-“) . . . (q-+1 - q-m-l), (qn - q-n)(qn-1 - ql-“). . . (q - q-1)
1, 0,
m > n > o n=O
otherwise.
Mathematically, U,(g) has the structure of a Hopf algebra with co-product, and antipode defined, respectively, as
co-unit
A(!$‘) = ,F1 @ lCfr, A(e) = iii @ a + a @ I+,
E(kf1)= 1, S(k”)
= Icy,
a = ei,
fi,
(20)
c(fi) = 0,
6(ei) =
S(a) = -q-%qhp,
for a =
(21) ei, fi,
(22)
where p is the half sum of positive roots of g and h, is determined by p(hp) = (,u,p), Vp E H*. These mappings are extended uniquely to all elements of U,(g) such that A and E constitute algebra homomorphisms and S is an anti-automorphism. U,(g) also carries the structure of a Hopf algebra with the opposite co-product z and antipode 3 (cf. equations (14) and (15)). This structure is equivalent to replacing q by q-’ in equations (20) and (22). The algebra U,(g) contains two Hopf subalgebras U+!(g) and U;(g) generated by {ei, Ic”}~zl and {fi, k”}~zl, respectively. Let {e,} be a basis of U;(g) and {es} be the corresponding dual basis for U,‘(g). By the quantum double construction [4], U,(g) is quasi-triangular with R E U,(g) ~3 U,(g) given by
with inverse R-’
= (I ~3 S-l)R
= C e, ~3 S-‘(eS).
Then u = C S(es)e, and u-l =‘C S-2(es)e, s s P(u)
= ZLuu-1,
satisfy Va E Uq(g).
(23)
On the other hand, equation (22) implies S2(a) = q-2hPaq2hP,
Va E U,(g)
(24)
so that [5] c=u
-1q-2hp
E centre of U,(g).
When acting on a finite-dimensional irreducible representation takes the eigenvalue [5] x4(C) = q(“J+2p).
of highest weight .4, C (25)
102
J. R. LINKS and M. D. GOULD
Using this invariant, we can obtain characteristic polynomial identities and generalized Gel’fand invariants for U,(g). Throughout we let ~4, be a tied but arbitrary finite-dimensional irreducible representation. Let us define a matrix A E U,(g) 6x1End V(Ao) by A = (q - q-l)-l(I
@ I - C-’
8 rn,(C-‘)8(C)),
(26)
where d is given by equation (9). A can be considered as a matrix with entries from U,(g). When A acts on a finite-dimensional irreducible module V(A), it satisfies the polynomial identity
fitA- w(A)0
=
o,
(27)
i=l
where ai
= (q _ q-t)-r(1
_ *(X”Xi+2”+2p)-(“o,no+2f))
and {Xi}{=t is the set of distinct weights in V(Ao). position of the tensor product space is V(A) @ V(Ao)
= 6
(28)
To see this, suppose that the decom-
miV(Ai),
i=l
where rni is the multiplicity of V(A,). XA,(A)
= (4 -
When A acts on V(A,),
q-‘)-1(1
-
it takes the eigenvalue
x~(C-‘>X~,(C-‘)XA,(C>).
Using equation (25) and the fact that each Ai is of the form Ai=A+xi,
with X, some weight of V(Ao)
we have
Qi(A) = Xx,(A) = (q _ q-1)-1(1 _ q(X*,X~+2”+2p)-(“o,Ao+2p)), Equation (27) then follows. Note that as q --f 1 we have ai
which is identities different Since
+ $(Ao, A0 + 2~) - $(Xi, Xi + 211 + 2~)
(29)
the familiar result for the Lie algebra polynomial identities [13]. Thus our are a generalization of these; we note that the matrix A and identities (27) are to the ones previously obtained in ref. [9]. A centralizes a(u,(g)) we obtain the family of Casimir invariants Cc”) = rq(Am),
where TV= r+, which is given by equation (10). Here we have chosen for convenience to take q-traces rather than u-traces (cf. equations (23) and (24)) although they both lead to equivalent sets of invariants.
103
CASIMIR INVARIANTS FOR HOPF ALGEBRAS
When acting on a finite-dimensional value
irreducible module V(A), C’(m) takes the eigen-
(30) where ni is the multiplicity of Xi in V(.4,) and dim, V(A) is the q-dimension of V(A); i.e. dim, V(A) = tr .rm(q-2hp). The q-dimensions have been determined explicitly [lo] and are given by
dim,v(n) = II
q(“+PP)
_ q-(“+P>4
q(p,o)
_
q-(p,“)
(31)
’
CrEQ
where 9 denotes the set of all positive roots of the underlying Lie algebra g. The proof of the eigenvalue formula for equation (30) is identical to that of Gould et al. [9] (see also ref. [12]). Unlike the invariants of reference [9], the invariants above reduce in the q + 1 limit to the invariants constructed by Gould [12] for simple Lie algebras which includes the Gel’fand invariants of the classical Lie algebras as a special case. The matrix A defined by equation (26) can in fact be determined explicitly. Note that equations (19) and (22) imply that A(C) = (C @ C)R%. Thus A is also given by A = (q - q-l)-‘(I
@ I - RzoR,,,),
where R$
Rn, = (I @ TA,)R,
= (I @ nn,)RT.
A formulation for R has been obtained by Kirillov and Reshetikhin [17] which reads R
=
,cb, hi@h’
where K = i(dimg Ln]
q
=
- T),
Q”- q-” q_q-’
’
qp =
qw*MPMP))
[nlq! = [nlq[n - llq.. . [llq, r
where a+)
is defined by [hi, Ep] = (ai, a(p))Ep
and hi satisfies 2 p(hi)v(hi)
= (p, v)
i=l
for all p, v E H’. The Ep and Fp are defined in ref. [17] and we refer the reader to that paper for details. The formula for R allows us to then determine A which is
104
J. R. LINKS
and M. D. GOULD
given by
@ rAo(E;n’ . . . E,mK) 1
We remark that since that representation 7rn0 is finite-dimensional, the sums over (2) and {m} truncate to finite sums. The matrix A defined by equation (26) is not the only matrix which gives us an analogue of the characteristic matrix. If we define the matrices A’ = (q - q-1)-1(1@ B = (4 -
B’ =
q-‘)-ye
(q - q-‘)(C
I -
C-l @~A,(c-l)s(c)) = (q - q-y(I@
c3 7rn,(c)a(c-‘) -
8 ~n,(C)a(c-‘)
I -
RnoR;o),
I @I) = (q - q-‘)-‘(R;;,‘(R-‘);o
- I CSJ I) = (q -
q-l)-l((R-l);oR$
- I @ I),
-
I CZJ I),
where 8 is defined by B(a) = (I @ 7Q,)&-L) = c cJ2) @ 7r~&(‘)), (a) then A’ satisfies the same identity (27) as A while B and B’ satisfy
n(B -
Pi(A)Q = 0,
i=l
where Pi(A) is given by pi(n) = (q _ q-l)-‘(q-(X”Xi+2”+2p)+(Ao,Ao+2P) _ I). Consequently, these matrices will also allow us to obtain sets of Casimir invariants. The matrix B lies in the centralizer of a(U,(g)) so we obtain a set of invariants by D+)
= TV.
When acting on V(n), these invariants take the eigenvalues (32) i=l
These eigenvalues are clearly different from those given by equation (30) so we must conclude that {C(“)} and {DC”)} are two different sets of invariants. However as q 4 1,
CASIMIR INVARIANTS FOR HOPF ALGEBRAS
105
xn(C(“)) and xn(D(“)) both re d uce to the same eigenvalue formula, so the two sets of invariants are equivalent in this limit. The matrices A’ and B’ both lie in the centralizer of a(U,(g)). Under the co-product a, U,(g) has a Hopf algebra structure with antipode 3 = S-l. Thus the square of the antipode is given by -2 s (a)
=
s-2(a)
=
2b (
q2h%q-
v’a E Udl?).
This allows us to construct the sets of invariants C’(“) = T&(A’)“),
D’(“)
= q_l((B’)y,
where T~.-~ E rg2,,,. It is not hard to deduce that the eigenvalues of C’(“) are given by equation (30) while those of D’(m) are given by equation (32), bearing in mind that the q-dimensions are invariant under q + q-’ (equation (31)). Thus we can conclude that C(“) = C’(4, D(“) = D’(m). Using these results we are now in a position to determine the images of these invariants under the action of the antipode S. Let A{ denote the highest weight of the dual representation of TA,; i.e. r/l;(e)
= 7r&(S(a)).
We let ‘;i:be the matrix obtained by replacing XA, with 7r,r(;in the definition of A, and we let C@) denote the corresponding R$RA,
=
C
et%
invariants. We then have @
7TAo(etes)
t,s = c
etes @I ?r~$S-‘(etes))
= (S o I)(RA;R~.)~.
t,s
It follows that A” = (S 18 I)((Ti’)m)t
A = (S @ I)(A’)t,
from which we obtain C(“) = T,(A”)
= (I@ tr)(l@
= (s ‘8 I)(1 8 tr)(l = S(T,&iy)
r&-2hp))(s
@II)((SI’)m)t
@ 7rAb; (q2hp))(A’)m
= SpQ).
Similarly, we also have the result for the invariants arising from the matrix B; i.e. DC”) = S(D(“)). To conclude this section, we briefly mention how projection operators can be constructed from the matrix A (or A’, B, B’). Acting on V(A), we have that A satisfies equation (27); i.e. h(A i=l
- cQ.4)1)
= 0.
106
J. R. LINKS and M .J. GOULD
Let {y,(A)}:=,
be the maximal set of distinct ai
and let P, be defined by
( A-y&v
p, = fj
).
%(A) - -d4
s=l
s#r It is easily verified that these operators satisfy the relations t
PA
c
= &tPt,
P,=I@I
T=I
and P,. projects onto the irreducible representations of the tensor product space with highest weight A + A,. These projection operators play an important role in obtaining solutions to the Yang-Baxter equation [4, 61. 6. U,(gl(n)) invariants
and polynomial
identities
In this section we will apply the general theory we have developed to U,(gl(n)). For this case we are able to obtain generalized Gel’fand invariants by taking q-traces of the matrix A = (q - q-‘)-‘(I
@ I - R; _&j-i))
and its powers. Here we have chosen the contragredient reference representation, which is defined by Q,-l)(a)
= &$Y(e)),
where y is the principal anti-automorphism generators of U,(gl(n)) by -/(JQ*1)
=
vector representation
(33) as our
a E U,(gl(n)), of U,(gl(n)). r(qkEii)
-J%,i*1,
y is defined on the elementary =
q~&i
and is related to the antipode by -/(a) = qhp s(a)q+. In order to determine
A explicitly we observe that
R =
c
e, ~3 es=
s
c
r(e,)
C3r(es).
.9
Thus
R= R(l.6)= (I @“(1,6))
c de,) @n(l,b)(y(es))= (Y 8 V$6,_1,. s
The matrix Rc1,6) has been obtained by Jimbo [7] and is given by
(34)
107
CASIMIR INVARIANTS FOR HOPF ALGEBRAS
where zji
(q - q-‘)q1/2(E~~+E3j-1)Ejji, qEjj,
=
i # j . . 2=j
and Eji is defined recursively as jSkSi.
Eii = EjkEki - q_lEkiEjk,
Here, eij are the standard generators of gl(n) in the fundamental vector representation. From equation (34) we then have
or
j>i
where we have used the fact that r*(u) = a, Va E U,(gl(n)). induction that T(&j)
1
=
q-(Pt-‘i-E3)~ij,
i
<
q(PyEi--Zj)kii,
i
1 j,'
It is easily verified by
j
(35)
-
where
Eii =
-_‘~ - q-‘)q-‘/*(E”“+Ejj-‘)E~i,
i z j
Q
2=j
. -7
.
and Eij is defined recursively as Eij = E,(,Ekj - qEhjElk,
isksj,
%*1
=
Ei,i*l*
In equation (35) we have used the standard basis for the weights of U,-,(gl(n)) satisfying (Ei,
Ej)
=
bij
and p is given by p
=-
1 n *C( n + 1 -2i)Ei. i=l
The matrix Rc6,_1) is then given by
R(lll-1)
=
f:
q(Pt’r4gi
@
eii.
(36)
j>i
By hermiticity we can deduce that R$ _l) is given by
Rg,-l)=cq-_(P7ej-Ei)Eji j
@
eii
(37)
108
J. R. LINKS and M. D. GOULD
and so substituting equations (36) and (37) into (33) we get that the entries of the matrix A are given by Aij = (4 _ 4-i)-i
(bij _ 2
q(P’2C~-i’-“)EieEkj)
k=iVj
=
c4 _
q-l)-l
(sij
_
2
+i+j-2k)&jjkj),
k=iVj
where a V b = max(a, b). It is easily checked that as q -+ 1, Aij -+ Eij, the generators of gl(n), so we have an analogue of the characteristic vector matrix of gl(n). Acting on an irreducible module V(A), A satisfies the polynomial identity fi(A
- cq(A)I)
= 0,
i=l
where &(A) = (q _ q-‘)-i(l
_ q-W”+“-i9
and Ai is given by LIP= (A, Q). This polynomial identity allows us to construct projection operators in the following way. First note that if A is a dominant weight then the w(A) are distinct; i.e. ai
= q(A)
=+ i = j.
Thus we can construct the operators P, =
satisfying n P,Pt = ws,
c r=l
P, = I@ I.
To determine the generalized Gel’fand invariants, we first note that (a~g,_l)(Q-2hp))ij = SijQ(n+1-2i)eii. The invariants are then given by Cc”) = rq(Am) =
C i,j,k . . . . 7
q(“+1-2i) AijAjk . . . A,i , \ m terms
which reduce to the usual gl(n) Gel’fand invariants as g -+ 1. Acting on an irreducible module V(A), these invariants take the eigenvalues
109
CASIMIR INVARIANTS FOR HOPF ALGEBRAS
Substituting equation (31) for the q-dimensions and using the fact that for gl(n), every a E @ is of the form CY= Ei - Cj,
i
we arrive at
XA(@))= (q -
q-1)-m
?(I
_
q-2(A;+n-i))m
fi
i=l
j=l j#i
q(At-Aj-l+j-i) q(Ai-A,+j-i)
_
q-(Ai-Aj-l+j-i)
_
q-(“i-“j+j-i)
.
We remark that in the limit q --f 1 the above formula reduces to the usual one for the eigenvalues of the gl(n) Gel’fand invariants as required. Finally, we note that the entries Aij of the matrix A span a submodule M of U,(gl(n)) under the adjoint action. In fact, the operators Aij transform as an adjoint tensor operator [i.e. as (1,6) 8 (0, -l)]. A n invariant bi-linear form which is non-degenerate on A4 may then be set up according to (-&j, AM) = tr~~,,0~(q-2hPAijA~~). Utilizing the construction of Proposition q2nC(2) = q2”
1 we arrive at the quadratic invariant
2
q(n+‘-2i)&Aji.
i,j=l
7. Conclusion In this paper we have developed some methods for constructing invariants for Hopf algebras. It was shown that there is a class of Hopf algebra (referred to as trace-type) which always admit invariant bilinear forms and provide a means of generating families of invariants. Due to a result of Drinfeld [5], it can be seen that all quasi-triangular Hopf algebras fall into this class. Using our results we were able to determine generalized Gel’fand invariants and characteristic polynomial identities for quantum groups. Their eigenvalues and characteristic roots have been computed and they are expressed in terms of the highest weights of the irreducible representations. These invariants and identities are not the only possible ones (see [9]) but we feel that they will be the easiest to work with because of the simplicity of their construction and eigenvalue formulae. We have’ also determined the images of these invariants under the antipode. To illustrate our results we have studied V,(gI(n)) in detail. Appendix A
In this appendix we derive the result (cf. equation (19)) A(&)
= u-’ 8 u-lRTR
E D(A) @ID(A),
where D(A) is the quasi-triangular Hopf algebra arising from Drinfeld’s quantum double construction [4]. Drinfeld showed that any Hopf algebra A may be naturally embedded
110
J. R. LINKS and M. D. GOULD
in a quasi-triangular Hopf algebra D(A). Let {e,} be a basis for A with multiplication, co-product and antipode given by T eset = mater, S(e,) = &t, A(es) = ,$et 8 e,, (38) where summation is taken over repeated indices. Let A0 denote the algebra dual of A, with corresponding dual basis {es} satisfying eSet = &rer,
A(eS) = rnFrer@ et,
S(e’) = (v-‘)ie”.
(39)
A and A0 can be identified as subalgebras of D(A) G A @ A0 in a natural way. We identify a E A with Z = a @ I and b E A0 with 3 = I @ b. The elements Z,, ES obey the relations (38) and (39), respectively. Multiplication between the elements {e,} and {es} is defined by - -t = e,@e’, e,e es& = ei m&m~ip~pjnr(V-l)Hej
@
= m~,m~i~:“~jr’(v-‘)lQejei. Drinfeld [4] has shown that the element R = Z, 63Z? satisfies RA(a) = &(a)R, (A 8 I)R = RnR23,
a E D(A) (I ~!aA)R = &d&2
(40)
and is invertible with inverse given by R-’ = S(E,) @ES so that D(A) is quasi-triangular.
It follows that 21= S(Z)&
$(a) As a consequence D(A) [18]:
= 21au-i,
and u-i = S2(Z?)F,
a E D(A).
satisfy (41)
of equation (40), the following result can be deduced to hold in ~~qm&&q
= p~qm~ti$.
(42)
Using equations (2) and (41) we conclude that A(S2(u)) = 218 uA(a)z~-’ @ u-l. Equations (42) and (43) will enable us to show A@-‘) = 21-l @ u-‘RTR. Indeed we have, by direct calculation A(&)
= A(S-2(i?)Q =2L -’ @ K’A(Z)~ @ 2~A(??s) from (42) =U -’ 8 ~-lm&Zr @Ftu @ uA(Z,) =u -’ @ u-‘Z
C% etu @ ~A(&&.)
=U -’ @ u-‘Z
@Ztu @ u&‘el @e,A(e,)
(43)
CASIMIR
INVARIANTS
FOR HOPF
ALGEBRAS
111
REFERENCES
PI PI [31 [41 PI PI [71 PI 191 IlOl PII WI u31 u41 PI WI u71 P81
R. J. Baxter: Exactly Solved Models in Statistical Mechanics, Academic Press, London, 1982. P. P. Kulish and E. K. Sklyanin: in “Integrable Quantum Field Theories”, Lecrure Notes in Physics 151, 61, Springer-Verlag, New York, 1982. E. Witten: Commun. Math. Phys. 121 (1989), 351; M. Wadati, T. Deguchi and Y. Akutsu: Phys. Rep. 180 (1989) 248. V. G. Drinfeld: Proc. I.C.M., Berkeley 1 (1986), 798. V. G. Drinfeld: Algebra and Analysis (in Russian) 1, No. 2 (1989), 30. M. Jimbo: Lett. Mafh. Phys. 10 (1985) 63. M. Jimbo: Left. Math. Phys. 11 (1986) 247. M. Rosso: Commun. Math. Phys. 117 (1988), 581. M. D. Gould, R. B. Zhang and A. J. Bracken: Generalized Gellfand Invariants and Characteristic Identities for Quantum Groups, University of Queensland preprint (Nov. 1989). R. B. Zhang, M. D, Gould and A. J. Bracken: Quantum Group Invariants and Link Polynomials, University Queensland preprint (Nov. 1989). M. D. Gould: J. Phys. A: Math. Gen. 20 (1987), 2657. M. D. Gould: Rep. Math. Phys. 27 (1989), 73. A. J. Bracken and H. S. Green: J. Math. Phys. 12 (1971), 2099; H. S. Green: J. Math. Phys. 12 (1971) 2106. M. E. Sweedler: Hopf Algebras, Benjamin, New York, 1969. E. Abe: Hopf Algebras, Cambridge University Press, 1980. L. D. Faddeev, N. Yu. Reshetikhin and L. A. Takhtajan: Algebraic Analysis 1 (1) (1989) 129. A. N. Kirillov and N. Yu. Reshetikhin: Commun. Math. Phys. 134 (1990), 421. N. Yu. Reshetikhin: Quantized Universal Enveloping Algebras of the Yang-Baxter Equation and Invariants of Links I, II, L.O.M.I. (Leningrad) preprint E-4-87, E-17-87.