http://www.ictp.trieste.it/~pub_offIC/97/101

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

ON MATRIX QUANTUM GROUPS OF TYPE An

Phung Ho HaiInternational Centre for Theoretical Physics,Trieste, Italy.

ABSTRACT

Given a Hecke symmetry R, one can define a matrix bialgebra ER and a matrix Hopfalgebra HR, which are called function rings on the matrix quantum semi-group and matrixquantum groups associated to R. We show that for an even Hecke symmetry, the rationalrepresentations of the corresponding quantum group are absolutely reducible and computethe integral on the function ring of the quantum group, i.e., on HR. Further, we show thatthe fusion coefficients of simple representations depend only on the rank of the symmetry,and give the explicit formula for the rank or 8-dim of HR-comodules. In the general case,we show that the quantum semi-group is "Zariski" dense in the quantum group. Thisenables us to study the semi-simplicity of the associated quantum group in some cases.

MIRAMARE - TRIESTE

August 1997

1 Introduction

Matrix quantum groups of type An generalize the standard deformations of the ma-trix group GL(n) introduced by Faddeev, Reshetikhin, Takhtajian [?] as well as multi-parameter deformations of the super-group GL(m|n) introduced by Manin [?]. In fact,to every Hecke symmetry R there is associated a bialgebra ER and a Hopf algebra HR,that are considered as the ring of regular functions on the corresponding matrix quantumsemi-group and group respectively. In the case when R is the Drinfel'd-Jimbo matrixof type An or its super analog, this Hopf algebra reduces to the above mentioned defor-mations. These quantum groups were firstly studied by Gurevich [?], who generalizedLyubashenko's results on vector symmetry for the case of Hecke symmetry.

Many interesting properties of quantum matrix (semi-) groups were found [?, ?, ?, ?, ?].These results show on one hand the similarity in the theory of quantum matrix groups oftype An and the classical groups GL(n) and GL(m|n), and on the other hand the relationsbetween quantum matrix groups of type An and the Hecke algebras.

In this work we study rational representations of matrix quantum groups of type An,i.e., comodules of the corresponding Hopf algebras. In case R is an even Hecke symmetryof rank r, we show that the category of comodules of R is equivalent to the category of rep-resentations of the standard quantum groups GLq(r) introduced by Faddeev, Reshetikhinand Takhtajian [?]. We show that this category is a ribbon category and explicitly com-pute the rank or 8-dim as well as the ribbon-dimension of simple comodules. Further, wegive the explicit formula for the integral on HR. This formula seems to be new even inthe case of GL(r).

According to our results, one can conclude that, over a field of characteristic zero,to each non-zero and non-root of unity q there exists one quantum general linear groupGLq(n), which is to be understood as an abstract abelian braided semi-simple category.The fusion coefficients of GLq(n) (to be understood as a category) do depend only on nbut not on q. Each even Hecke symmetry of rank n provides a functor from GLq(n) tothe category of vector spaces which is to be understood as a realization of GLq(n), andsuch a realizion exists for every n and q (provided that q possesses a square root). Animportant invariant of GLq(n) is the 8-dim which shows that GLq(n)'s for different q arenot equivalent. The formula for 8-dim on simple comodules is explicitly computed, whichis quite analog to Weyl's formula for rational simple GL(r)-modules.

For an arbitrary Hecke symmetry, we show that the matrix quantum group is "Zariski"dense in the corresponding quantum semi-groups, that is, in dual, the canonical homo-morphism ER —> HR is injective. This fact enables us to study the semi-simplicity ofHR in some special cases. Our technique bases mainly on the theory of Hecke algebras.

The work is organized as follows. In Section ?? we briefly recall some definitions ofthe bialgebra ER and its Hopf envelope HR. In Section ?? we study the case where R isan even Hecke symmetry. Section ?? is devoted to the general case. In the Appendix webriefly recall some results on Hecke algebra from [?, ?] and prove some lemmas needed inour context.

Let us fix a field K of characteristic zero. All objects of the paper are defined over K.

Throughout the paper we shall deal with an element q from K x which we will assume notto be a root of unity, but may be the unity itself.

2 The Matrix Quantum Groups and Quantum Semi-

groups

Let V be a finite dimensional vector space over K and R : V 0 V —> V 0 V be aninvertible K-linear operator. Fix a basis x1,x2,... ,xd of V and let ξ 1 ,ξ 2 , . . . ,ξd be itsdual basis in the dual vector space V* to V, < ΞI|XJ >= 5). Denote zji := ξ i 0 xj, thenzji, 1 < i, j < d form a K-linear basis of V* 0 V. Let i? - be the matrix of R with respectto the basis xi 0 xj, xi 0 xjR = xk ® xlRijkl. The algebra ER is defined to be the factoralgebra of the tensor algebra on V* 0 V by the relation:

mn k l p V q kl / 1 \1 hj ^m^n ^% ^] J hpq • \ ^)

Remark. The definition of ER is, in fact, basis free.ER is a bialgebra, the coproduct and counit are given by A(z*) = z\®zkj and ε(zji) = δj

respectively (we shall frequently use the convention of summing up by the indices thatappear both in lower and upper places). We shall be interested in right ER-comodules,the coaction of ER will be denoted by δ. ER is called a matrix quantum semi-group iff Rsatisfies the Yang-Baxter equation:

(R 0 V)(V 0 R)(R ®V) = {V®R){R® V)(V 0 R),

and matrix quantum semi-group of type An iff R also satisfies the Hecke equation:

for some q G K x (Kx := K \ {0}). In this paper we shall be interested in the latter case,i.e., the operator R is assumed to satisfy both equations. We call such an operator theHecke operator. We define the following algebras. ΛR and SR are factor algebras of thetensor algebra upon V by the following relations:

—xi 0 xj = xk 0 xlRijklkl, ^\

qxi 0 xj = xk 0 xlRijkl,

respectively. These algebras play the analogous role of the antisymmetric and symmetrictensor algebras on V.

The most well known example of R is the Drinfel'd-Jimbo matrix Rd given by:

Rd : = q T. eij ~ (Q+ D ( q ε ^ *' eij - q

where ε(ij) := sign(i — j). Rd is an even Hecke symmetry of rank d = dim( V ) . Theassociated Hopf algebra HR (see below) is the general linear quantum group GLq(d) [?].Analogously, the general linear quantum supergroup is given by the following matrix:

m+n m+n ε(ij) ε (

= qj2 4 - (q +1) v d r-A - (-i^V—FT4-)> (4)

where i is the parity of the (homogeneous) basis element xi. Rm|n is a non-even Heckesymmetry if m, n = 0. The associated Hopf algebra HR is the general linear quantumsupergroups GLq(m|n) [?, ?].

The matrix quantum group associated to R is defined to be the Hopf envelope of ER.By definition, the Hopf envelope of a bialgebra E is a pair (H, i) consisting of a Hopfalgebra H and a bialgebra homomorphism i : E —> H, satisfying the following universalproperty:

for any Hopf algebra F and a bialgebra homomorphism e : E —> F, there existsuniquely a Hopf algebra homomorphism h : H —> F, such that e = h o i.

This definition is dual to the definition of the maximal subgroup of a semi-group. Becauseof that we use this definition as the standard way of finding a quantum group from a givenquantum semi-group. The Hopf envelope always exists (cf. [?, ?]).

Let HR be the Hopf envelope of ER, then HR is, in general, infinitely generated as analgebra. However, if R is a closed Yang-Baxter operator, then HR will be, in fact, finitelygenerated. This fact was first proved in [?] with a slightly stronger assumption. Since weare most interested in this case, below we will briefly recall the construction of HR in thiscase.

Let R : V ® V —> V ® V be a Yang-Baxter operator. Then R induces a braiding inthe monoidal category generated by V and R, that is the category V, objects of whichare V®n, n = 0,1,2,..., morphisms are obtained from R and the identity morphisms bycomposing and taking the tensor product. In particular, the braiding on V ® V is R[?]. ER can be considered as the Coend of the forgetful functor from V to the categoryof vector spaces [?]. We want to extend V to a braided rigid category. The problem isto extend R to a braiding on V ® V* and V* ® V, here V* is understood to be the leftdual to V. The candidates of these operators can be the inverse of the half dual i?b of Rand (R~lf [?]. Let evV : V* 0 V —> k be the evaluation map, evV(ξ <g> xj) = 5), anddbV : k —> V ® V* be the dual basis map, dbV(1) = xi ® ξ i, then the half dual to R,i?b : V* 0 V —>V(g)V* is defined as follows:

i?b := (evV 0 V 0 V)(V 0 R 0 V)(V 0 V 0 dbV).

With respect to the basis xi 0 xj, i?b is given by

f 0 XiFt = X l ® ξkRjk

Thus the operator twisting V and V* can be given by:

with P satisfying R%P^ = δniδjm (and hence P%R\™ = δniδjm). It was shown by

Lyubashenko that if Ft is invertible then so is (i?"1)^ [?], and the operator twisting

V* and V is given by:

e ®xJRv*y = xl®eR-l%-

A Yang-Baxter operator R is said to be closed iff the operator Ft is invertible (andhence so is (R~lY). In this case the Hopf envelope HR can be characterized as the factoralgebra of the tensor algebra on the vector space, generated by zij ,tji, 1 < i, j < d, by thefollowing relations (cf. [?, ?]):

Rmnz k I _ V q pkl /r\

t)4 = si (6)

z% = 51 (7)

The coproduct is: A(z{) = zk ® z\, A(^') = tk ® t{, S(xij) = tji. To show that HR is aHopf algebra it is sufficient to find the antipode on tij. Multiplying (??) with tks from theleft and trj from the right and using (??), (??), we have

or equivalently

mn l j m p klij znltqj = tmk Zi Rpq

lk znltq — lk Zi Pnm

Setting n = j in (??) and summing up by this index we get Bkp = tmkzipBmi, or

tTB^B-ll = 51 (10)

where Bmi := Pl

mli. Analogously, setting k = p in (??) and summing up by this index, weget:

C^aCX = Si, (II)

where Cji := Pfv (??) and (??) imply that

S(tkj) = BlzlB-'l = C-ll

kzl

nCl (12)

Thus we showed that HR is a Hopf algebra, hence the Hopf envelope of ER.Equation (??) shows that V being a simple comodule on ER may turn to be reducible

being comodule on HR. In [?], there were introduced a condition ([?], Eq. (20)), whichwas equivalent to the following:

BC = const • I,

that makes the second equation in (??) trivial. It is easy to show (see Section ??) thatthis equation will be satisfied if R is a Hecke operator. In this case

BC=(q-l-(l-q-l)Tr(C))I.

In Section ?? we will show that if R is a Hecke symmetry, i.e., R is a closed Hecke operator,then simple ER-comodules remain simple being considered as HR-comodules.

5

3 Matrix Quantum Groups Associated with Even Hecke

Symmetries

If R is an even Hecke symmetry then there exists a one dimension ER-comodule whichyields a group-like element, called quantum determinant, in ER. The localization of ER

with respect to the quantum determinant is HR. We study in this section comodules ofHR in this case.

3.1 Simple Comodules of ER and HR

By its definition, ER is a graded algebra. Its n t h component En is the factor space of(V <S> V)®n. En is a coalgebra and every ER-comodule decomposes into the direct sumof En-comodules [?]. En, the dual vector space to En, is an algebra and acts on every(right) En-comodules from the left. In fact, we have an equivalence of the two categories:

EiM = ME".The Hecke algebra Hn (see Appendix) acts on V0n from the right via R, for x G V0n,

xTi : = xRi, where Ri : = idy<8>j_i ® R <S> idv®n-i-i, 1 <i < n — 1.

Thus, we have an algebra homomorphism

ρn : Hn -^ End(y®n), ρn(Ti) = Ri. (13)

Let us denote by Hn the image of Hn under ρn. It was shown in [?] that the actions ofHn and E^ on V®n commute and the following relations hold:

E* = E n d W n ( F ^ ) = E n d ^ ( F ^ ) , (14)

(V^n). (15)

Hn is semi-simple, in fact, it is the direct product of matrix rings over K [?, ?], hence En

and Hn are semi-simple. Let I := Ker(ρn), then Hn is isomorphic to the two-sided idealof Hn, {h £ Hn|hI = 0} and Hn decomposes into the direct product of algebras:

= Hn x I .

Thus, any primitive idempotent of Hn belongs either to Hn or I . By virtue of (??), everysimple module of En is a submodule of V®n considered as an E^-module. Let L be asimple En module and i : L —> V®n be the isomorphism identifying L with a submoduleof V®n, then i splits. Let p : V®n —> L be an ^-comodule homomorphism, such thatpi = idL. Under the isomorphism in (??), ip corresponds to a primitive idempotent inHn. Conversely, let l be a primitive idempotent in Hn, then Im(l) is an E^-module, forif h G Hn,x G V0n, h((x)l) = (h(x))l. Moreover, this ^-module is simple, by virtue of(??). Thus, a primitive idempotent in Hn yields a simple module of En.

Let 1 = J2 ei be the decomposition of the unit in Hn into central minimal idempotents,then, as an Hn-module, V<Sm is decomposed into:

Therefore E^ decomposes into the direct product of its minimal two-sided ideals, whichare themselves algebras:

Hence, a simple E^-module is the simple module of some E*ni := EndHnIm(ei), for someuniquely defined i. Therefore, two primitive idempotents in Hn define isomorphic simpleE^-module iff they belong to the same minimal two-sided ideal of Hn. Thus we provedthe following

Lemma 3.1 Every simple En-comodule has the form

where l is a primitive idempotent ofHn. Conversely, every primitive idempotent l ofHn

induces a simple comodule of En iff p(l) =£ 0. Further, the modules defined by l and I' arenon-isomorphic iff l and I' belong to different minimal two-sided ideals.

Remark. Since Hn is the direct product of matrix rings over K, so is E*n. ThereforeSchur's lemma holds for ER - the automorphism ring of a simple ER-comodule is K.

A Hecke symmetry R is called even Hecke symmetry if R the algebra ΛR defined in(??) is finite dimensional. It was shown by D. Gurevich that in this case ER owns agroup-like element, called quantum determinant, and the localization of ER with respectto this element is a Hopf algebra. We want to show here that in this case the canonicalhomomorphism i : ER —> HR is injective. First, we need the following

Proposition 3.2 Let R be a Hecke operator. Let r be the largest number, such thatAr

R =£ 0. Assume that r < oo; then ΛR

r is a simple ER-comodule and for every simpleER-comodule N, ΛR

r ® N is simple too.

Proof. It is known (and easy to prove) (cf. [?]) that:

AR = Im(ρ(yr)),

where yr := 1 J2weer(~(i)~l<yW^w ( s e e Appendix). Thus, A^ is a simple ER-comodule,

since yr is a primitive idempotent. By the assumption of the Proposition we have:

^ 0 and ρ(ys) = 0, Ms > r + 1.

Let N be a simple ER-comodule of homogeneous degree m > 1. By Lemma ?? thereexists a primitive idempotent θ in Hm, such that A = Im(ρ(θ)). Ar

R<£>N is a homogeneouscomodule of degree r + m. Embedding Hr and Hm into Hr+m as follows:

"Kr3Tt^Tte Hr+m, l < i < r - l

Hm 3 T3 ^ Tr+j G Hr+m, 1 < j < m - 1,

we have Ar

R<£>N = Imρr+m(yrθ). Thus, it is sufficient to show that ρr+m(yrθ) is a primitiveidempotent in Hr+m. This is precisely the assertion of Lemma ??.

The number r in Proposition ?? is called the rank of R. A direct consequence of thisProposition is that A^.® ΛrR is a simple comodule. Hence, the braiding on A^.® ΛRr shouldbe scalar. Since the identity operator is not closed, except when dimkΛrR = 1, R is notclosed if dimkΛRr > 2 (cf. [?]).

Now assume that R is an even Hecke symmetry, then dimkΛrR = 1, and hence inducesthe quantum determinant denoted by D in ER. It is shown in [?], that ER[D~l] is aHopf algebra, hence coincides with HR (cf. [?]). Since HR is a Hopf algebra, every finitedimensional HR-comodule possesses the dual. Let Ar

R denote the left dual HR-comodule

to Λr

R then

as HR-comodules, by definition of a dual comodule and since the antipode sends a group-like element to its inverse.

Theorem 3.3 Let R be a Hecke symmetry. Then the following assertions hold:

(i) The canonical homomorphism i : ER —> HR is injective,

(ii) HR is cosemi-simple,

(iii) Simple comodules of HR have the form N®Ar

R*®n, where N is a simple ER-comodule,n>0.

Proof. According to Proposition ??, N <S> r

R is simple whenever N is. Hence, D is nota zero-divisor in ER. Thus, the localizing homomorphism ER —> ER[D~l] = HR isinjective, (i) is proved.

Let M be a finite dimensional comodule on HR = ER[D~1], and xi,i= 1,2,... , m b ea K-basis of M, then there exist elements aij, 1 < i, j < m in HR, such that δ(xi) = xjaji.Let n be such an integer, that aijDn E ER,\/i,j. Then M ® AR

0n is an ER-comodule.Let M C N be finite dimension HR-comodules. There exists n EN such that M®Ar

R®n

and A Λ Ar

R®n are ER-comodules. Hence, there exists ER-comodule P, such that:

iv ® 1\R = 1V1 ® 7VJJ © -T.

Thereforeiv = M © _r R A^

Thus every finite dimension HR-comodule is absolutely reducible. Since every HR-comoduleis an injective limit of a system of finite dimension comodules, (ii) is proved. (iii) alsofollows from this discussion.

In Section ?? part (i) of Theorem ?? will be generalized to the case of arbitrary Heckesymmetry. But we emphasize that Proposition ?? holds also for arbitrary (non-closed)Hecke operator.

8

Since minimal central idempotents of Hn are indexed by partitions of n, to any par-tition λ of n there associates a comodule Mλ which is either zero or simple comodule ofHR.

3.2 The Integral on HR

By definition, a (left) integral on a Hopf algebra HR is a linear functional / : H —> ksatisfying the following condition:

x) = ^2xi 1 (x2). (16)(x)

If there exists an integral which does not vanish on the unit of HR, then HR is cosemi-simple. In this case we can rescale /, such that /(I) = 1 and call it a normalized integral.It is known that if a normalized integral exists then uniquely [?]. Our aim is to find suchan integral for HR.

Let us make HR into a Z-graded algebra setting degZ (zij) = 1, degZ (tij) = — 1. For xhomogeneous, A(x) = xi®X2, then degZ(x) = degZ(x1) = degZ(x2). Hence, (??) is nota homogeneous relation unless degZ(x) = 0, while HR is homogeneous. Thus, we provedthe following.

Lemma 3.4 The value of J on a homogeneous element of HR (with respect to the abovegrading) is equal to zero unless the degree of this element is zero.

The problem reduces to finding the values of / on monomials in zji's and tkl's, in whichthe numbers of zji's and if's are equal. By virtue of the relation in (??), every monomialin zji's and tlk's can be expressed as a linear combination of monomials in which zji's areon the right of tkl's. Hence it reduces to finding:

ni l ,32,— ,3n,h,h,-.. ,ln f - 7 7 7T . ( 731 732 73nfhfh Jn \Ii1, i2,...,in,ki,k2,...,kn " / 1 z »2 in tk1tk2 ' ' t kn)

J

At first sight, it is not clear if any set of values of f on zij1zi j • • z^ntlttlL 2 • • tln inducesan integral on HR, since there may be more than one way of expressing a monomial in z'sand t's as a linear combination of zij1zi j • • zijn tl1 t • • tlu . Here the Yang-Baxter playsan important role, which ensures that different ways of expressing yield the same result.In fact, it can be shown that there exists a bialgebra structure on ER ® ER

OP,COP such thatHR is isomorphic as a Hopf algebra to a factor of ER ® ER

OP,COP by some biideal (cf. [?],see also [?]). In this construction ER is generated by zij and ERop,cop by tji, the biideal isgenerated by (??) and (??).

According to (??), we have:

' mm m raPl>P2'"' >Pn,h,h,-.. ,ln rail ,32,--- ,3n,h,h,... ,ln

By virtue of (??), this is equivalent to

For fixed i1,i2,... , in, h, h, . . . , ln, (??) is a relation on z's being considered as elementsof HR. By Theorem ?? ER is injectively embedded in HR, hence (??) is a relation for z's

n

as elements of ER. By virtue of (??), (??), I, for fixed i1, i2, . . . , in, h, h, . . . , ln, shouldn

represent an element of Hn. Thus, I can be written in the following form:

31,32,•••,3n

where Rw = ρn(Tw) as in (??), {Λw |w & Sn} is a set of matrices indexed by w E Sn.n

According to relation (??) on zji, we have the following relation on Λw (we omit indicesfor simplicity):

2 J n w Ri <S> Rw = 22 nw ®RiRw, 1 < i < n — 1. (19)

Let us denote εi(w) : = 21(l(w) — l(viw) + 1), where vi = (i,i + 1) is a basic transpositionin Sn, w E Sn. We have

RiRw = qεi(w)Rviw - εi(w)(q - 1)Rw.

Thus, (??) can be rewritten as:

nw (Ri - εi(w)(q - 1)) - q1-*™ A , J 0 Rw = 0.

This equation would imply that

nw (R - £i(w)(q - 1)) - ql~£*{w) Λnviw= 0, (20)

if the Rw, w E Sn were linear independent being elements of Hn, in other words, if ρn in(??) were injective. This is, unfortunately, not always the case. Nevertheless, we assume

that Aw satisfy (??) which is equivalent to

Aw=q~l{w) ARw-i, A:=Ai .

Whence (??) has the form:

On the other hand, the relations (??) and (??) raise up the following relations on I:

. njl,J2,...,jn,h,h,-,ln • n-P1'^2'- dn-l,h,h,-,lnX*nj _ X3" Tl 1 Ii1,i2,... ,in,k1,k2,... ,kn

uk\ I i1,i2,... ,in-i,k2,ks,... ,kn ,

, njl,J2,...,jn,h,h,-,ln , n-P1'^'- Jn-l,l2,h,-,ln

C XT = C1 Tjn Ii1,i2,... ,in,k1,k2,... ,kn in I i1,i2,... ,in-l,k2,k3,... ,kn .

10

Substituting (??) into (??), we get

n-pl,j2,... ,jn-l,h,h,... Jn-il

For w £ Sn there exists unique w1 E &(^n-iti) (the Young subgroup corresponding to thepartition (n — 1,1) of n) and n > k > 1, such that w = wn-k • • • vn-2Vn-\W\. Consideringw1 as an element of &n-i, we have, for k < n — 1,

j1,j2,...,jn

,•••/? \P™,---,P2r> jl,J2-,3n-l-k jrin-l)kn,...,k2

WlPn,--.,P3,p2

1

and for k = n — 1,31,32, - j n TV//^^ R 32,---,3n

For an operator Rw = Rn-k- • • Rn-2Rn-\Rw1, 0 < k < n — 1,w1 E (5(ra_i):L) (k = 0means w E &(n-iti)), set

Rn-k • • • Rn-2RWl if k > 1; ,^-,

cRwl if k = 0 l (

and extend it linearly to a map Trq,c(Rw)n : H n > 7~tn-i- Then we have, for c = Tr(C),

/ <fcl o 31,32,••• ,3n m / o \njl,32f-,3n (0f\\Jn i^Ti)'" ;"'2)"'l ^ ' ^ ^ kji,--- jk2jfC\ ^ ^

The following property of the operation Trnc,q follows immediately from its definition and

l 0ifρn(h) = 0. (27)

We emphasize that (??) would not be true if R were not closed. We see that two evenHecke symmetries induce the same operator Trq iff they are of the same rank.

Let c = Tr(C), then the right-hand side of (??) is equal to (indices are omitted)

A R^ ® T^ldRw) = E Tr(C)q-lW A Rw-i 0 Rw

W۩(n_i>i)

Z_^ Q A Rw-lRn-l ' ' ' Rn-k <8> Rn-k ' ' ' -Rra-2-Rw

In the left-hand side of the equation one should have in mind the identity &n-\ = &(n-\ti),thus in the sum, w~l runs in &(n-\ti) while w runs in &n-\. Let us first examine the second

11

It

term of the right-hand side of this equation for K fixed (omitting Λ for the moment):

q (-w-) Rn_k • • • Rn_iRw-i ® RwRn_2 • • • Rn-kE

E q (w> Rn_k • • • Rn_2Rn-iRw-i ® RWVn_2Rn-3 • • • Rn-k

Rn_k . . . JRra_2JRra_ lJRra_2JRw_1 0 Rw(Rn_2)2Rn_3 • • • Rn_k

E Q

E <rwee

E

(n-1,1)

Rn.-2Rn.-lRw-1

-Rra-fc k - - Rn-l(Rn-2)-\-qRn-iRn-k • • • Rn-2R

R'n-'n-2

® Rwvn-2Rn-3

w-1 <8> RwRn-3

<-Wl'n ^ Rn-lRn-k • • • Rn-2Rn-\RWVn_2 <8> RWVn_2Rn-3

~ <-W->~ Rn-\Rn-k • • • Rn-lRw-1 ® RWRn-3

/ Q Rn-2Rn-lRw-1 <8> RwRn-3 ' ' ' Rn-

(repeating the above process k — 2 times)

Rn-lRn-2 ' ' ' Rn-kRn-k+1 . . . Rn.-2Rn.-lRw-1

Thus, denoting

Ln : = q Rn-l + q Rn-2Rn-\Rn-2 + ••• + (? n Rn-lRn-1

L1 := 0, and c := Tr(C), we have

' R1R2 • • • Rn-l,

n>2,

E v~l E (28)

Remark. The above manipulation also applies for Tw e H n which yields (??) for Rw

replaced by Tw.Also (??) has the following form:

ra-l

( Λ (c -i 0

Thus, if Rw,w G @n-i are linear independent, then we can conclude that

ra-l

A (29)

12

0

Since we assume that /(I) = 1, we have Λ= 1. Hence, provided that Lk + c, 1 < k < nare invertible, we have

+ c)-1. (30)k=1

Remark, that Lk is the image of the Murphy operator Lk through ρ (see Appendix), theycommute since the Murphy operators commute.

Let R be an even Hecke symmetry of rank r, then c = — [—r]q [?]. Hence (c + Ln) isnot invertible in Hn for n > r. Fortunately, Lemma ?? shows that (c + Ln) is invertible

n

in Hn, thus (??) always has a meaning. It remains to check the relation (??) for I. Byvirtue of (??) and (??), (??) has the form:

E n-l{w) Hn \( A ^An\(n _L T \ ~l R -, R 31,32,••• ,3n

q Q] W A C2D0)(c + Lini riw-i\ t\wh b b

According to (??), we have:

. . .

Thus, (??) can be rewritten as:

X <S> R w = ^ q~l{w) ™A Rw-i ®

Using the method of the proof of Equation (??) we can show the following equation:

\ "• n — l(w)rpn frp \ „ rp \ ^ n~ l(w)rp „ rp / . j \ / on\

Let p E Hn such that p = ρ(p) satisfies p(c + Ln) = 1Hn. Let

(c + Ln)Tv = E cvuTu and pTv =

thenC^ UJW±\U (JUtViXv.

Set < Tu,Tv >= δu,vql(v) and extend it linearly on Hn, thus we get a non-degenerate

scalar product on Hn which satisfies < hk, g >=< h, gk* >, where * is the linear extensionof the map Tw i—> Tw-i,w £ Sn ([?], Lemma 2.2). Since (c + Ln) is *-invariant, we have:

l(w~) W ,- rji / I T \rJ1 -^. ^ - / I 7" \rT1 rf1 ~~~~^

y C j \ -i-w) \J~-' ~r J-^n)-*-v -^ ^ \^ ~r ^n) -*- w ) -*-v -^ •

13

Hence

Ln)Tw= ]T ql^-l^cwv%. (33)

Since Tw,w E Sn form a basis for Hn, we can conclude that cuv = ql^~1^cvu. Applying

operation * on (??) we get

Now, we have

E 9"l(

we&n

q

«€©„

(by virtue of (??)) = ]T g - ^ ^ - i ) 0 ^ (c + X^.ueen

Since (c + Ln) is invertible, (??) is proved. Thus, we have

Theorem 3.5 Let R be an even Hecke symmetry of rank r. Then the integral of HR canbe given by the following formula

I= E q-l{w)(\{{T~k + c)-l\c®nRw-,®Rw. (34)Sn k=1 )

3.3 Structure of HR

We now study the fusion rules for HR-Comod. Let A h n and JJL\- m and Mλ and MM bethe associated simple comodules of ER, and hence simple comodules of HR. The tensorproduct of Mλ and M^ decomposes into the direct sum of simple ones:

where c{^ denotes the multiplicity of M γ entering the above decomposition. c{^ arecalled fusion of Clebsch-Gordan coefficients. Our aim is to show that these coefficientsdepend only on the rank of R. Let eλ and eM be primitive idempotents of Hn and Hm

respectively, Mλ = pn(e\),Mn = Pmie^). Embed Hn into H n + m , T i—> Ti and Hn intoTj i—> Tj+n. Then MX®M^ = Im(pra+m(eAe™)), where e™ is the image of eM under

14

the defined above embedding. The decomposition of eAe™ into primitive idempotents

depends only on λ and JJL themselves. In particular, let λ = (λ1λ2,... , λk), then we have

the decomposition

M ( λ 1 ) 0 M ( λ 2 ) 0 • • • 0 M(Afc) = ]T m v M M , (35)

with the coefficients m\^M^ as in the case of representations of GL(n). They are given

by the formula (cf. [?], Section 6)

Hence we have the determinant form for M λ (cf. [?], Section 19 ):

M A - d e t (M{Xi_i+j))i<ij<k . (36)

Hence the problem reduces to investigating when ρn(eλ) = 0.

It is known that ER and HR are coquasitriangular bialgebras (cf. [?, ?]). Let τ

denote the braiding on HR-Comod induced from the coquasitriangular structure on HR.

In particular, ryy = R, TV*,V = Rv*,v, Tvy* = Rvy*- Let M be an object in HR-Comod,

we define its 8-dim to be [?]:

8-dim(M) := evMrM,M*dbM(1K).

For example 8-dim(V) = Tr(C). More general, for a morphism f : M —> M we define

its 8-Tr to be

8-Tr(f) := evMTM,M*(/ 0 M*)db M (l K ).

Then we have

8-dim(M) = 8-Tr(idM), (37)

8-Tr(fg) = 8-Tr(gf). (38)

Lemma 3.6 Let M be a simple HR-comodule, then 8-dim(M) = 0.

Proof. Since M is simple,

Hom(M* 0 M, k) ^ Hom(M, M) = k. (39)

dbM : k — > M 0 M* is not trivial, hence induces a morphism p : M 0 M* — > k, such

that p o d b M = 1k. Thus p = 0, hence O ^ p o r M J M * : M* 0 M — > k. By (??) there

exists c E kx, such that p o r M | M * = evM, hence evMτdb = c ^ 0.1

Let M = M1 © M 2 with morphisms pi : M — > Mi, ji : Mi — > M, pi o ji = 1Mi,

j1p1 +j2p2 = 1M. Then

8-dim(M) = dbMi"M,M*(l 0 (j1p1 + J2P2))EVM

= E i dbMTM,M* (1 0 jipi)evM

= E i dbMi"M,M* (jt* ®Pi)evMi

= Ei dbM(Pi ®3t)TMi,M;evMi

= dbM i((j1p1 + j2p2) 0 l)TM i,M;evM i

= 8-dim(M1) + 8-dim(M2).

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Thus the 8-dim is additive.Let us fix the following standard rule of defining the dual to the tensor product. Let

Vi, i = 1, 2,... , k be HR-comodules, then

(Vi 0 V2 0 • • • 0 Vk)* := Vk* 0 Vk*_x 0 • • • 0 V7,

with the evaluation map ev = evfcevfc_i 1 •-evi. Let M = Im(ρ(e)) be a simple HR-comodules, e is a primitive idempotent in HR. Considering e as a morphism of HR-comodules, since e = e2, we have

8-dim(M) = dby«m(e <g> e*)TV®ny®

= dbyigm(e ® V®n*)Tv®™ y®n*evy(gm(lfc) . .

= 8-Tr(Ve) ' [ V

where wn is the longest element of Sn, that reverses the order of the elements 1,2,... , n.The last equation above can be best explained using braid representation of τ (see theAppendix). Thus we can consider 8-dim(M) as an element of H0 acting on V®° = k.

By induction we can express wn as wn = wn-\vn-\ 1 • • v2v1, hence we can easily checkthat (Twn)

2 = qn-\l + (q- 1 ) ^ ) 7 ^ . Therefore

fl . (41)k=1

Hence, assuming that e = ei,λ, λ h n as in Lemma ??, (ii), and using (??), we have

8-dim(M) = g - s ( A ) - r a ( r a - 1 ) / 2 T r ; ) C ( T 4 ( - • • (Trq,

cn(e)) •••)), (42)

where s(λ) := Y^k=\ rλ(k) (see Appendix, Theorem ??), which depends only on λ but notthe λ-tableaux.

Proposition 3.7 Let R be an even Hecke symmetry of rank r. Let λ h n and M =Imρn(eλ). Then Mλ = 0 (and hence is a simple comodule of HR) iff λr+1 = 0.

Proof. By virtue of Lemma ??, it is sufficient to show that

Tr;;r

2'-'ra(e) := T r ^ T r 2 , ^ - • • (Tr" r(e)) • • •)) ^ 0

iff λ r +1 = 0. Let Rr be the Drinfel'd-Jimbo R-matrix of type Ar-i, defined in (??), thenRr has rank r. Let ρnq denote the representation of Hn induced by Rr. If λ r +1 = 0 thenV®n, being considered as Hn-module, contains direct summand isomorphic to the simpleHn-module Sλ (see Appendix) ([?], Proposition 5.1). On this module eλ = 0, henceIm(ρn(eλ)) = 0. By Lemma ?? and (??), Trq,r1,2,...,n(eλ) = 0.

If λ r +1 = 0, then V®n, being considered as Hn-module through ρnq, does not containdirect summand that is isomorphic to Sλ as Hn-module, so that eλ acts as zero on V®n.Hence TrJ;2'""ra(eA) acts as zero on V®° = k, by (??). On the other hand, TrJ;2'-'ra(eA) isa constant, hence it is equal to zero.

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Theorem 3.8 Let R be an even Hecke symmetry of rank r. Let λ h n, JJL h m, then wehave

c^M,. (43)

γr+1=°

Hence, the category HR-Comod is equivalent to the category of representation of the stan-dard quantum group GLq(r).

3.4 The 8-dim of simple comodules

The 8-dim is an invariant in HR-Comod, however it is not tensor multiplicative and henceis difficult to compute for non-simple comodules. Even the formula for 8-dim(l/0ra) is verycomplicated. We provide in this section the so called ribbon-dimesion - rdim - which ismultiplicative and, on simple comodules, it differs from 8-dim only by a power of q. Infact, for a simple ER-comodule M of degree n, defined by a primitive idempotent e G Hn,its ribbon-dimension is, by definition,

rdim(M) :=Trq,c1,2,...,n(e). (44)

Since ER-Comod is semi-simple, one can extend (??) linearly on all ER-comodules. Toshow that rdim is well defined, one has to verify that the right-hand side of (??) isindependent of the choice of the idempotent e. Before showing this, let us make it clearthe meaning of ribbon-dimension.

By definition, a ribbon in a strict braided category is a natural isomorphism θ, subjectto the following equations:

n (n ,QS o WUMigiN — θ M Vy θ N ) i ,

9M* = (9(M)Y,

where τ denotes the braiding in the category. This definition first appeared in the worksof Joyal and Street [?] and Turaev [?]. A strict braided rigid category with a ribbon iscalled ribbon category. For more details on ribbon categories, the reader is referred toa book of C. Kassel [?]. For an object M in a ribbon category, the ribbon-dimension,rdim(M), is by definition the morphism

evMTM,M* (θM 0 M*)dbM : I —> I. (46)

By virtue of (??), rdim is multiplicative (cf. [?], Chapter XIV).

Lemma 3.9 (i) ER-Comod possesses a ribbon. Let e be an idempotent in Hn, andM = Im(ρ(e)), then the ribbon on M is given by

θM := Twn2e. (47)

(i) θ extends to a ribbon on HR-Comod, hence HR-Comod is a ribbon category.

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Proof. (i) follows from the fact that Twn

2 commutes with all elements of Hn. This can bemost easily seen from (??) in which Twn

2 is a symmetric polynomials on L1, L 2 , . . . , Ln,hence belongs to the center of Hn according to Theorem ??, (i). One can also verifydirectly that Twn

2Ti = TiTwn

2.(ii) follows from Theorem ??, (iii). One sets 6\r* = (#AO* and uses ( ? ? ) .

Corollar 3.10 (??) is well defined and rdim is tensor multiplicative.

Proof. Let e1,e2 £ Hn define the same comodule, then they belong to the same minimaltwo-sided ideal of Hn. Hence they are conjugate, e2 = ae\a~l. Thus, we have

2,...,n(e2)= 8-Tr(Twn

2e2)

= 8-Tr(a" 1 T W n

2 ae 1 )

= 8-Tr(T w n

2

e 1 )

According to (??), we have the formula for the rdim of Mλ:

rdim(Mλ) = det ( r d i m ( M ( A . _ i + : 0 ) ) i < . ^ . (48)

Direct computation shows that

^ * « - " F = i w - (49)

The proof of the following proposition will be given in the Appendix.

Proposition 3.11 Let Mλ be a simple ER-comodule corresponding to a partition A h n .Then

— o TT [A, - λ j + j - i]

[J-

Corollar 3.12 With the assumption of Proposition ??

8-dim(Mλ) = g- i ( r a 2 +£I=i^+ 2 '

In order to extend the formula (??) to a formula for all HR-comodules we need thefollowing lemma.

Lemma 3.13 Let M,N be ER-comodules, and N* be the left dual comodule to N, then

8-dim(M <g> N*) = evN®MTN®M,M*®N* {{TM,NTN,M)2 ® M* ® N*)dbN!S)M- (52)

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Proof. The proof is based on the fact that N is isomorphic to the left dual of N* by meansof the brading. In fact, we can choose ev v* = TN,N*£VN, db v* = db/-^— ~l

Let A = (Ai, A2,... , Ar), Ai G Z, Ai > A2 > . . . > Ar. Let Ao := Min{Ai}, A* := \ - Ao.Define

Mλ : = M X ® A r 0 A°,

where λ = (λ1, λ 2,... , λ r), and

Ar®x° if λ0 > 0;

A"®|AO | otherwise.

Theorem 3.14 Let λ and Mλ be as above, then

]J [λi ~ λ j + j ~ i]q, (53)

where n = Y<K-

Proof. If λ0 > 0 then (??) is just (??). Assume λ0 < 0. According to Lemma ??, it issufficient to show that

By means of the equality Tjjy®w2 = {TU,V®W){V®TJJ>W2){TU,V®W) it is sufficient to check

the above equation for the case n = 1. In this case, rsry1 = RrRr-\ 1 • • R1R1R2 • • • Rr =qr(1 + (q- 1)Lr). We have Λ r = Im(ρ(y r)) and qryr(1+ (Λ,Vq - l)L~r) = 1qr+11, that proves

the equation above. Equation (??) can then be derived by easy computation.

We see that 8-dim is an invariant of the category HR-Comod up to a scalar constant.In fact if we make the transformation R 1—> k • R, which also deforms the Hecke equation,then 8-dim will be multiplied by k~n , where n is the homogeneous degree of the comodule.In particular we have

= 8-dim(Λr) = g " ^ + 1 ) / 2 .

Further, we remark that Mλ ® Λr and Mλ, A h n , have different 8-dim:

8-dim(Mλ 0 Λr) = g- ( r + 2 r a ) ( r + 1 ) / 2 8-dim(M A ).

Assume now that the quantum determinant commutes with elements of HR (see [?]for the necessary and sufficient condition). Then we can set it equal to 1 to get the specialmatrix quantum group SHR. In this case, the comodule Λr, being considered as a SHR-comodule, is isomorphic to the trivial module. Hence, it is expected that this module has

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an 8-dim equal to 1. This can be made by rescaling R i—> q ^ . We call the rescaled8-dim the normalized 8-dim and denote it by 8-dim,

\ 1 —U

[j - i]

The last equation shows that the 8-dim of Mλ and of Mλ ® Λr are equal. As a corollaryof the above discussion we have

Proposition 3.15 Let R be such an even Hecke symmetry that the quantum determinantcommutes with all elements of HR, SO that one can set it equal to 1 to get the correspondingspecial matrix quantum groups SHR. Then the coquasitriangular structure < -\- > on SHRcan be defined, setting

< zk|zl >= q-r-^Rlk

4 Non-Even Hecke Symmetry

We want to show in this section that the canonical homomorphism i : ER —> HR isinjective if R is a Hecke symmetry, i.e., a closed Hecke operator. Then, using the methodin Section ?? we give the criterion for non-semi-simplicity of HR for non-even Heckesymmetry R.

Let V be the K-additive category, generated by V and R, that is, objects of V areV®n and morphisms in V are obtained from R and the identity morphisms by composing,taking tensor products and linear combinations with coefficient in K. Analogously, letW be the K-additive monoidal category, generated by V, V* and RV,V, Rv* ,V, Rvy* andevV, dbV. It is known that ER (resp. HR) is isomorphic to the Coend of the forgetfulfunctor v : V —> Vectk (resp. w : W —> Vectk) (cf. [?]). Relations on ER (resp. HR)are obtained from morphisms in V (resp. W). In fact, the relations on ER are obtainedfrom Homy(l/®ra, V®m). To show that i : ER —> HR is injective, it is sufficient to showthat

Homv(T/0 r a, V®m) ^ H o m w ( F 8 " , V®m). (54)

If m / n, then both sides of (??) are empty, hence assume that n = m. Since R satisfiesthe Yang-Baxter equation, one can represent RV,V, Rvy*, Rv*,V by oriented braids:

10010.4ex1010RV,V := T T .4exl010Ry)y* :=

The morphisms evV and dbV are represented by oriented semi-circles:

603.4ex1010evV := * .4ex1010dbV := T

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