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Classification of finite dimensional unitary irreps for U q [gl(m‖n)]M. D. Gould and M. Scheunert Citation: Journal of Mathematical Physics 36, 435 (1995); doi: 10.1063/1.531317 View online: http://dx.doi.org/10.1063/1.531317 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/36/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Matrix elements and duality for type 2 unitary representations of the Lie superalgebra gl(m|n) J. Math. Phys. 56, 121703 (2015); 10.1063/1.4938076 Classification of unitary and grade star irreps for U q (osp(2‖2n)) J. Math. Phys. 36, 531 (1995); 10.1063/1.531321 Finite dimensional irreducible representations of the quantum supergroup U q (gl(m/n)) J. Math. Phys. 34, 1236 (1993); 10.1063/1.530198 Classification of all star irreps of gl(m‖n) J. Math. Phys. 31, 2552 (1990); 10.1063/1.529001 Classification of all star and grade star irreps of gl(n‖1) J. Math. Phys. 31, 1524 (1990); 10.1063/1.528695

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Classification of finite dimensional unitary irreps for &[gNml @I

M. D. Gould Department of Mathematics, The University of Queensland, St. Lucia QLD 4072, Australia

M. Scheunert Physikalisches Insitut, Universittit Bonn, Nussallee 12, D-53115 Bonn, Germany

(Received 27 December 1993; accepted for publication 5 August 1994)

All finite dimensional irreducible unitary representations of the quantum supergroup f-JqliK4~)l are classified in terms of their highest weights, for generic q>O. 0 1995 American Institute of Physics.

I. INTRODUCTION

The theory of Lie superalgebras (LSAs) and their representations plays a fundamental role in the understanding and exploitation of supersymmetry in physical systems. The concept of supersymmetry first arose in elementary particle physics’ and has since been applied in a variety of other areas, including condensed matter physics2*3 and nuclear physics.4,5 A major new develop- ment in the theory and applications of LSAs came with the introduction of quantized LSAS,~-* commonly called quantum supergroups, which are (nontrivial) Z$graded extensions of quantum groups.‘,” These remarkable algebraic structures give rise to a universal R-matrix which affords a systematic construction of solutions to the Yang-Baxter equation of importance in the quantum inverse scattering method,” exactly soluble lattice models,” and the theory of Knots and Links.13-15

In applications such as these, it is inevitable that those representations (reps) of quantum supergroups of physical interest will be the unitary reps which are the natural generalization of unitary reps of simple Lie (super) algebras.16 Such finite dimensional representations share many of the fundamental properties of finite dimensional reps of a semisimple Lie algebra and in particular are completely reducible. In Ref. 17, it was shown that all highest weight integrable irreducible reps (irreps) for a quantized KacMoody algebra, which includes the finite dimensional irreps of a quantum group as a special case, are equivalent to unitary it-reps. However, this is certainly not the case for quantum supergroups even in the classical limit.‘8T19 It is our aim in this paper to systematically classify all finite dimensional unitary irreps for the quantum supergroup U,[gl( m In)] for generic q>O.

Aside from their physical importance, such irreps are mathematically interesting as well. For example, in Ref. 20 it is shown that for a given finite dimensional unitary irreducible U,[gl(mln)] module V, that the braid group generator arising from the universal R-matrix is diagonalizable on V@ V, regardless of multiplicity. This affords a spectral decomposition in terms of central projec- tion operators for the braid group generator analogous to that found by Reshetikhin13 and Gould2* for quantum groups. Such a spectral decomposition is an important first step in attacking the Baxterization procedure.‘0T22 Moreover link polynomials arising from this approach afford new insight into the systematic construction of two-variable link polynomials. This arises because type I quantum supergroups, and U,[gl(mln)] in particular, admit one parameter families of (nonequivalent) unitary h-reps so that their corresponding link polynomials automatically contain an extra parameter.23

The paper is set up as follows: Sec. II is introductory while in Sec. III we introduce the definition of unitary representation (two types) and investigate some general properties. In Sec. IV we outline the connection of our formalism with the super-Hermitian forms and corresponding

0022-2488/95/36(1)/435/18/$6.00 J. Math. Phys. 36 (l), January 1995 Q 1995 American Institute of Physics 435

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436 M. D. Gould and M. Scheunert: Unitary irreps for U,[gl(mln)]

super conjugation operations of Ref. 20, of fundamental importance to the diagonalization of the braid group generator arising from the universal R-matrix of U,[gl( m 1 n)]. In Sec. V a criterion for unitarity based on the eigenvalues of the “second order” Casimir invariant of U,[gl(m In)] is given. This result is applied in Sec. VI to give a complete classification, in terms of highest weights, of the finite dimensional unitary irreps. We conclude in Sec. VII with a brief summary of our main results and some suggestions for future work.

II. QUANTUM SUPERGROUP U,(mjn)

Throughout U,(mln) denotes the quantum supergroup U,[gl(m]n)] with standard generators

e,=E,,+1 v fCl=E*+hl (l=Sa<m+n), qt(“2)EaQ (1 GaCm+n), (1)

where the E,, ( 1 s a s m + n) form the usual basis for the Cartan subalgebra H of gl( m In). The defining relations of U

4 (mln) will not be given here and we refer to Refs. 6-8 for details. We

merely note that U,(m n) has the structure of a Z,-graded quasitriangular Hopf algebra24 with coproduct

and antipode

*tq (1/2)E,,)=q(1/2)E,,~q(1/2)E,, 7

A(x)=qha’2@x+x@q-ha’2; x=e,,fa, (2)

S(u)=q-hpy(u)qhp, uE U,(mln), (3)

where y is the principal antiautomorphism defined by

y(l)= 1, y(x)= -x; x=e, da .E,,

and extended to a Z,-graded algebra antihomomorphism in the obvious way so that for homogeneous U,U E U,(mln),

y(uu)=(-l)[~l[~ly(u)y(u),

where [u] E Z2 denotes the parity ‘*,i6 of u E U (m In). Above, p denotes the graded half-sum of the positive roots of gl(m]n) and h,, h, (l~a<qn+n) are defined by

W,)=@d, Uha)=(A,qJr AEH*,

where { LY,}~~~- ’ is the set of simple roots and (,) is the natural invariant bilinear form induced on H”.

To clarify the above notation it is convenient to introduce even indices iJ= 1,. ..,m and odd indices ,u, u= 1 ,...,y1 in terms of which the Cartan basis elements may be expressed Eii (1 Siam), E,, (IGpsn). The corresponding dual elements 4, S,EH* defined by

Ei( Ejj) = Sij , Ei(Epp)=Oy

6p(Eii)=O, ~,(LJ= a,,,

form a basis for H”. The invariant bilinear form (,) induced on H” is then defined by

(EirEj)=Sij, (sp,S,)=-Sp,, (EirSp)=(Sp)Ei)=O.

Following Kac,25 as a set of simple roots we choose the distinguished set

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M. D. Gould and M. Scheunert: Unitary irreps for U&gl(mln)] 437

LTi=Ei-Ei+l (lGi<m); cY’,=e,-61; a/.L= sp- a,+ 1 (lSP-+),

where cr, is the odd simple root. The corresponding sets of even and odd positive roots of gl(m]n) are given, respectively, by

We denote the half-sum of the even and odd positive roots resp., by po, pt so that the graded half-sum of the positive roots is p=po-pt (see Ref. 18 for fully explicit results).

We may likewise express the quantum supergroup generators (1) in the dual index form

ei=Eij+ 17 fizEi+li 9 hicEii-Ei+ Ii+* 9 lSi<m (44

e,=E,,i, fs=Eim 9 h,=E,,+Eii

ecc=E /-&CL+ 1’ fp=E++l+, h.=E,+1,,+1-E,,, lsp+, (4b)

where i denotes the odd index p= 1. These generators give rise to a (&-consistent) Z-gradation of UJmln):

U4(mln)=t13U~)(mln), n

with the definition (cf. Kac, Ref. 25)

deg(ei)=deg(f;) =deg(e,) =deg(f,) =deg(~,,) =o,

&de,) = - degOc,) = 1.

It is worth noting that the generators (4a) form the generators of the quantum group UJ m) = U,[gl( m)] which, in view of the coproduct rule (2), forms an even Hopf subalgebra of U4( mln). Similarly the generators (4b) constitute those of the even Hopf subalgebra V,(n) = C/,[gl(n)] although in this case the coproduct (2) gives rise to the opposite coproduct on U,(n) (obtained by replacing q with q-’ in the usual definition). The generators (4a) and (4b) collectively form the generators of the reductive quantum group

U,[gl(m)~gl(n)l~~U,(m)~uU,(n).

The universal R-matrix for UJmln) is even and has the following general form

(6)

where the u, (resp. us) are linear sums of monomials in the q’a”, f, (resp. qha’*, e,); for fully explicit results see Ref. 26. The R-matrix is the unique (up to multiplication by a central factor qxkhk@‘i; h, , h; E H) element of the above form27*28 satisfying the relations

Ar(u)R=RA(u), VUE U,(mln), (84


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where we have adopted the standard notation.7’8’24 In particular, AT= ToA is the opposite coproduct where T:U4(m~n)@U4(m~n)+U,(m~n)@UU4(m~n) is the twist map defined for homogeneous elements u, u E UJ m ] n) by

We note that lJ,(mln) also constitutes a Z,-graded quasitriangular Hopf algebra under the opposite coproduct with antipode S-’ and universal R-matrix

RT=T(R)=z u’@u,. s

We now say something about the representation theory of U,(mln). Every finite-dimensional UJmln) module V admits a Z,-grading (compatible with the supergroup grading)

V= Vi@ Vi,

where V,j (resp. Vi) is referred to as the even (resp. odd) component of V. We then say that u E V is homogeneous if u E VtjU Vi and define the parity factor [v] =O (resp. 1) according to whether u E VG (resp. Vi). The finite dimensional irreducible U,(mln) modules are uniquely labelled by their highest weights A and have the same dimensions and weight spectrum as the corresponding gl(mln) modules of the same highest weight.29 We denote the set of dominant weights of gl(mln) [and hence U,(mln)] by D,. For AED+, we let V(A) denote the irreducible UJmln) module with highest weight A and we let rrA be the representation afforded by V(A).

Throughout, V,(A) denotes the irreducible U,(m)@U,(n) module with highest weight AED, and we denote the set of distinct weights in V,(A) [resp. V(A)] by II,(A) [resp. II(A)]. Following Kac,25 we say that A e D + and the corresponding irreducible UJ m In) module V(A) are typical if

(A-I-p,cu)#O, tla+;

otherwise, A and V(A) are called atypical. We note that every finite dimensional irreducible U,(mln) module admits a natural

Z-gradation

(9)

compatible with the Z-grading (5) on U,(mln), in which case we say that V(A) admits d + 1 levels [V,(A)#(O) assumed]: here V,(A) is the Z-homogeneous subspace consisting of vectors of degree -k. The Z-gradation (9) then induces the following partitioning of the weights in V(A):

W=k~oWU (10)

where &(A) denotes the set of distinct weights in V,(A). This latter space is to determine a module over the even quantum subgroup U,(m)@ U,(n) from which it follows that II,(A) is stable under the Weyl group of gl(m) $gl(n). We assume, without loss of generality, that V,,(A) belongs to the even subspace of V(A).

Corresponding to each finite dimensional irrep rr* we have a Casimir invariant (Ref. 30)

(11)


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M. D. Gould and M. Scheunert: Unitary irreps for U,[gl(mln)] 439

where str denotes the usual supertrace. On an irreducible finite dimensional module V(p), p E D + , the above invariant takes the constant value3’

x,(c,) = C (- l)[ilniq2(“+P-xi), (12)

where the sum on i is over the distinct weights Xi, each occurring with multiplicity ni in V(A), and [i] denotes the parity of weight Xi : In the notation of Eq. (lo), we have (- l)t’] = ( - l)k, for Xi E n,(A).

To be more explicit, let (~2 denote a weight basis for V(A) with basis vector u, having weight A, and parity [cz] E Z2. We find it convenient to introduce the matrices [notation as in Eq. (7)l:

Nd,s= 2 (- 1 )[s1[p3G4(rp~s, CR;flap= 2 (- 1 )[S1rP3Q4ap~s, (13) s s

where [s] = [u,] = [us] is the parity of U, , u”~U,(mln). The invariant (11) may then be expressed in terms of the above matrices according to

c,=c (- 1)[‘Y]q2(A.,‘P)[R~R*],,, a

01’)

where matrix products are defined in the usual way so that

[R:kJap=~ [R~la,[R~l,p~ Y

We conclude this section by noting that the Z-gradation defined by (5) induces the following decomposition of the universal R-matrix (7):

-deg(u,)=deg(us)=n

R= 2 R(n), R(“)= c U,63US. (14) ll20 s

It follows in particular that R(O) can be composed only of elements of the even quantum subgroup (6). For elements u of this latter algebra we have, by equating Z-graded components of AT(u)R=RA(u), that

A’(u)R(O)=R(~)A(U), Vu E U,(m)@UJn).

Moreover by equating zero Z@Z@Z-components of the triple tensor product equation (8b), it is easily established that R(O) satisfies the relations

It then follows, from the uniqueness of the R-matrix for quantum groups,27*28 that

R(‘)=R(m)@R(n), (15)

where R(m), R(n) denote the universal R-matrices of the quantum groups U,(m), U,(n) c U,(m)@ U,(n), respectively.


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III. UNITARY REPRESENTATIONS

Following Scheunert et a1.,16 the Lie superalgebra gl(m]n) admits two classes of unitary irreps, which have been classified by Gould and Zhang. 18*19 These two classes are referred to as type (1) and type (2) *-irreps in Ref. 18. Below we alter this terminology a bit and refer to them as type (1) and (2) unitary irreps. Our aim in this paper is to extend these results to the quantum supergroup U,(mln). Throughout, we assume that q>O is a positive real parameter.

In the notation of Eq. (l), we let t be the conjugation operation on Ug(mln) defined by

which we extend to an (even) algebra antihomomorphism to all of U,(m In) so that

(UU)t=UtUt, vu,u E U,(mln).

We note that t so defined is indeed consistent with the ~~(rnln) defining relations* as well as the coproduct; i.e.,

A(u~)=A(u)~, VUE U&FZ[~), (16)

where (u@u)~=( - l)tU1tU1~t@~ t, for homogeneous U,U E U,(mln). [We stress that this rule defines a conjugation operation on the graded tensor product U,(mln) @ U,(mln) whereas the usual prescription (U @ u) t = u + @ u t does not.] Moreover, if y denotes the principal antiautomorphism on U,(mln), we have

rw)=M41t. (17)

From Eq. (3) we thus have, under the action of the antipode S,

s(ut)=[S-l(u)]?

We say that a finite dimensional module V is unitary of type (1) [resp. (2)] if it can be equipped with an inner product (,) satisfying

(u+u,w)=(u,uw) [resp. (-l)r”l(u,~w)]

Vw,u E V, where [u] is the parity of homogeneous u E UJmln); equivalently, if T is the representation afforded by V,

T(u+)= 9-rt(u) [resp. (- l)fU1rrt(u)],

where t on the rhs denotes normal Hermitian conjugate. Following Ref. 18, the type (1) and (2) unitary irreps are related via duality. We define the

dual irrep rx of rrA, AED+, by

m;(u) = &+4L (18)

where T denotes supertranspose, so that in a homogeneous basis for V(A) we have, for homogeneous u E U,(m[n)

T$(u),p=( - l)[“‘wT*(u)p,.


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M. D. Gould and M. Scheunerf: Unitary irreps for U,[gi(mln)] 441

Note that this definition of dual representation differs from the usual one which here would have y replaced by the antipode S. However, in view of Eq. (3), both definitions give rise to equivalent representations. We note that if rA is a unitary irrep of type (1) [resp. (2)] then rrf is unitary of type (2) [resp. (1)] and conversely.

Indeed, suppose T,, is unitary of type (1) so that

Tr~(u)=~*(ut).

Then, the corresponding dual irrep. satisfies

~X(U)t,B=7TX(U)B,=~~(Y(U))P~=(-1)[U1[a’~a(;/(u)),p,

where the overbar denotes complex conjugate. Using Eq. (17), together with the fact that (for a nonzero matrix element) [u] =[ol]+[p] (Mod 2), we obtain

++)t,p=(- 1) [Uwr*(y(u)+)p,= (- l)[Ul([al+[P1)rr~(y(u+))ap,

or

so that 7~: is unitary of type (2). Conversely the dual of a type (2) unitary irrep is unitary of type (1).

Denoting the irreducible module dual to V(A) by V*(A), we have thus proved: Proposition 1: For A ED + , V(A) is an irreducible unitary module of type (1) if and only if

V*(A) is an irreducible unitary module of type (2). Cl The task of classifying the unitary irreps is facilitated by noting that there is a natural method

(Ref. 18) of inducing nondegenerate invariant sesquilinear forms on V(A) from a given irreducible unitary U,(m)@ U,(n) module VO(A): recall (Refs. 17 and 32) that all irreducible finite dimensional modules for a quantum group are equivalent to unitary ones. Let us therefore assume that V,(A) is a unitary irreducible U,(m)@ V,(n) module, so that V,(A) is equipped with an inner product (,) satisfying

(utu,w)=(u,uw), tlu~U,(m)~U~(n);u,w~V~(A).

Note that this automatically implies that A ED + must be real. We then extend this form to all of V(A) by defining [notation as in Eq. (9) and cf. Eq. (5)]

(V,,(A),V,(A))=O, O-GWd,

(u+u,w)=( - l)e(u,uw), \du E Uk’)(mln), (19)

(cru~+pu~.w)=Ly*(u*,w)+p*(u~,w), Va,pEC.

Here, &z& is a grading parameter leading to two inequivalent forms. Then it can be shown that (,) determines a (well-defined) sesquilinear form with the properties

(u+u,w)=(-l)[U]B(u,~~);(~,~)=(~,~), \Ju,w~v(A), (20)

for arbitrary homogeneous u E U,(mln). As for gl(mln) (Refs. 18 and 19), U,(m)@ U,(n) modules with different highest weights are orthogonal under the form (,). In particular the decomposition (9) is orthogonal with respect to the form (,). The proof of these results is virtually identical to the classical case” [see in particular Lemma (1) of Ref. 181 and will not be reproduced here.


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442 M. D. Gould and M. Scheunert: Unitary irreps for U,[gl(mjn)]

Following Ref. 18, we call any sesquilinear form on V(A) satisfying the conditions of Eq. (20), invariant of type 8. An important property of the above induced form is

Lemma 1: The form (,) induced on V(A) is the unique (up to scalar multiples) nondegenerate invariant sesquilinear form of type 13 on V(A).

Proofi See Lemma (2) of Ref. 18. 0 The induced form (20) has all the properties of an inner product except that it is not generally

positive definite. When it is, V(A) must be a unitary module of type (1) [resp. (2)] when 8=0 (resp. 1). In view of the uniqueness of the above induced form, it follows therefore that V(A) is a unitary module if and only if V,(A) is a unitary U,(m) ~3 U,(n) module and the corresponding form induced on V(A) is positive definite (and hence gives rise to an inner product).

Remarks: It is worth noting that all finite dimensional unitary modules are completely reducible. In particular the tensor product of two irreducible unitary modules of the same type is unitary, under the naturally inherited inner product, and thus decomposes into a direct sum of unitary irreducible modules of the same type. However, the tensor product of an irreducible type (1) unitary module with a unitary irreducible module of type (2), is not generally completely reducible (cf. classical case’8*‘9).

IV. SUPER CONJUGATION RULE FOR THE R-MATRIX

From the point of view of diagonalization of the braid group generator and link polynomials arising from quantum supergroups,” ’ ’ it 1s convenient to investigate the above results in the framework of super-conjugation operations and super-Hermitian forms.

A super conjugation operation * on a Z2-graded algebra A is a mapping A--+A with the following properties:

(9 [al = [a*], (ii) (aa+pb)*=&a*+pb*, ‘dcu,fl~C, (iii) (ab)*=( - l)‘al[blb*a*, (iv> a**=a, for all homogeneous a,b EA.

Note that condition (i) simply states that * is parity preserving. In the case of U,(mln) we have two such super conjugation operations defined by

u*= i

u+, [u]=O, CUP, [Ul’l, (21)

where E= ti is a square root of - 1 and t is the conjugation operation of Sec. III. In many ways superconjugation operations are more natural to work with. This is particularly the case since the universal R-matrix, Eq. (7), satisfies the important super-conjugation rule2’

R”=RT, (22)

where (u @ u) * 3 u * @ u *, \du, u E U4( m In). Moreover under the action of the coproduct we have

A(u*)=A(u)*, VuE U,(mln).

It is worth noting, since R is even, that the conjugation rule (22) is equivalent to Rt = RT. Continuing this approach, a nondegenerate sesquilinear form (,): VX V-+C on a &graded

vector space V = Vi $ V, is called super-Hermitian if

(i) (w,u)=O, for [wl#[u], (ii) (w,u) = ( - l)[wl[ul(u,w), \du,w E V,UVi.


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M. D. Gould and M. Scheunert: Unitary irreps for U,[gl( mln)] 443

The restriction (,& (resp. (J,) f o such a form to VG X Vi (resp. Vi X Vi) is Hermitian (resp. skew-Hermit&n); the latter simply means that i(,>, is Hermitian.

We moreover say that (,) is definite if and only if (u, u) #O, Vu # 0 E V. This implies that both (Jo, i(,)l are (positive or negative) definite. Throughout we assume that the restriction (Jo of such a form to Vi X V,- is always positive definite (else replace (,) with -(,)) in which case we call (,) a positive definite super-Hermitian form.

With this notation, we thus call an irreducible UJmln) module V= V(A) unitary if it can be equipped with a positive definite super Hermitian form satisfying

(,*U,W)=(-l)‘ulrul(U,UW), VW,V EViUVi, (23)

where * is a superconjugation operation (21) on U,(m In). Here, we demonstrate that this definition of unitarity is equivalent to the definition given in the previous section.

At first sight, it appears that there are two types of such unitary irreps corresponding to the two possible super conjugation operations (21). However, if V(h) is unitary under the form (,) with respect to the conjugation operation

u*= i

u+, [u]=O, eu+, Cul=l,

then it is easily seen that V(A) is also unitary with respect to the conjugation operation

u*= i

u+, [u]=O, -al+, [ul=[111

under the form

(v,w)‘=( - l)[~I[Wl(v,w).

It thus suffices to work throughout with the superconjugation operation (21) with E af?xed square root of -1.

Now let V= V(A) be a unitary U4(mln) module so that it is equipped with a positive definite super-Hermitian form (,) satisfying Eq. (23). Recall that the restriction (& of (,) to V, X V,j is positive definite while if (,)1 is the restriction to Vi X Vi then Ed is either positive or negative definite. We have (notation as Sec. III)

Proposition 2: V(A) is equivalent to a type (I) [resp. (2)] unitary module if and only if - E(,), is positive (resp. negative) definite.

Proof Suppose -c(,)~ is positive definite. We then have an inner product on V(A) defined by

We now demonstrate that V(A) is a type (1) unitary module with this inner product. Indeed for homogeneous u E UJmln) we have, in view of Eq. (21), that

(U+v,W)=((-E)~~lU*v,W), [U]E&

=(E)qU*v,W)

=(e)[“l( - E)[wl(u*v,w)

=(~)[~l(-~)[~I(-1)[~I[~l(v,UW)

=( e)[“l( - E)[“l( - l)[~~[“~(,)[~l(v,uw).


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For [u]=O we may assume that [w]=[v]=O or 1 so that

(u+v,w)=(v,uw).

For [u]=l we may assume either [v]=O and [w]=l or [u]=l and [w]=O; in either case we have

This proves that V(A) must be a type (1) unitary module. Similarly replacing --E with E in equation (*) above, it is easily seen that if E(,), is positive

definite then V(A) must be a type (2) unitary module. The converse holds by simply reversing the above argument. This is enough to establish the result. 0

Although for some problems2’ it is more natural to adopt the framework of super-Hermitian forms discussed above, for the purposes of classifying the unitary irreps, it is convenient to adopt the approach and conventions of Sec. III which we adopt throughout the remainder of the paper.

We conclude this section with a useful Hermiticity condition satisfied by the entries of the R-matrices (13). We have

w?&d+=C C-1) [S1[P1~*(u,)aa(u”)+=C (-l)[~l[~I(-.)[~l,~(,s)sn(u”)*. s s

Thus if rrA is a type (1) unitary it-rep., then

m la~)+=C C-1) [slq - e)4r*( l&,( us)*

=c (-l)[SIIPl+[sl,h(u,*)pa(uS)*=C (-l)[“l[“lrr*(~:)p,(u”)*=[R~]p,. s s

In view of Eq. (22), we thus arrive at the conjugation rule

(244

Similarly, if no is a unitary irrep of type (2), we arrive at

[R&?=( - l)l”l+I~l([R&J+. Wb)

V. INVARIANTS AND A CRITERION FOR UNITARITY

We shall be concerned here with the matrices (13) and corresponding invariants (11’) for the case 7~~ = rr’, is the fundamental (m + n)-dimensional defining representation. We note that this representation is undeformed and determines a two-level type (1) unitary irrep of gl(mln) and hence U,(mln). In view of the decomposit ion (14) of the R-matrix, we thus have

RE,=RLy)+RL:); Rt’=O, n>l.

For ease of notation, we drop the subscript E, and simply write R= R(“+R”) where R”),R”) denote the matrices [cf. Eq. (13)]

deg( u”) = e

R;z= 2 (- 1)lS1rb3,&s)ab~s; 8=0,1


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(1 ea,b<m+n) with a similar notation for RT. In dual index notation, we have

R ,E U(‘)(mln), CL’ 4

and note the following vanishing matrix elements:

R. =R!O)=R(q)=R!!‘=R(‘)=O [P 1P IL’ ‘I IJ‘V .

The corresponding invariant C = Cei of Eq. (11’) is given by

c= 5 q(2P*ei)[RTRlii- i q(2p*S~)[RTR]pG,

i=l p=1

which, from Eq. (12), takes the following eigenvalue

)/*(C) = 2 q2(A+P.ciL i q2(A+P,sp) i=l p=l

(25)

(26)

on the irreducible Uq( mln) module V(h). From Eq. (25), we may further write

m C= 5 q(‘P,‘l’RiT,Rgi+ C q n

(2p,Ei)R;;‘TRj:“- 2 q’2p4)R;;TRI0,’ 9

i=l i= 1 /&=I

which, in view of Eq. (15), simplifies to

C= v+ 5 qc2PTCi)R(m)zR(m)ji- c q ,a ‘2P*8~)R(n);$(n),,, i=l /.&=I

where

(2P,ei)RLRpj (27) i= 1

and R(m),R( n) denote the R-matrices corresponding to the defining (reference) irrep of the quantum groups Uq( m), Uq( n), respectively. Using p=po-p, , with’*,

PI=; g ;c sp3 ei- - t=l /.L=l

we may thus write

C= q+qpnC(m)-qvmC(n),

where C(m), C( n) denote the invariants corresponding to the defining irreps of U,(m), U,(n), respectively (with q replaced by q-’ in the latter).32’33

In the following, a fundamental role is played by the U,(m) 63 U,(n) invariant of Eq. (27) which may be alternatively expressed


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77= C-q-T(m) +4-V(n). (27’)

It follows that on an irreducible U,(m) ~3 U,(n) module V,(v) contained in an irreducible U,(mln) module V(A), h,vs D, , that 77 takes the constant value32333

i= I /.L=l

=x*(C)- 5 q2(v+P*si)- i q2(v+P38J

i

=x,(C)-~~(C),

i=l p=l I

with x*(C) as in Eq. (26). Now let n+(A), A ED, , denote the set of distinct U,(m)@ U,(n) highest weights occurring

in V(A). We then obtain the following criterion for unitarity (cf. Refs. 18 and 19): Proposition 3: V(A) is an irreducible type (1) unitary module if and only if

XA(C)-xv(C)=, ~v+AEII+(A). (28)

Proofi First, from Eq. (24a), since the vector irrep. is unitary of type (l), we have

qT= 5 q(2p4(Rpi)tRCLi. i=l

Thus if V(A) is a type (1) unitary module and u is a highest weight vector for a U,(m) 8 U,(n) module V,( v) C V(A), then we have

(2% i=l

which implies, since q >O, that x*(C) - x,(C) >O. Furthermore, equality occurs if and only if

R,iu=O, Vi,p,

which would imply, in particular, that

(here, as before, i denotes the odd index ,u= 1). Thus we would have

e,u=O,

for all simple roots cr, which can only occur if v is the unique maximal weight vector in which case v=A. This certainly shows the necessity of (28).

To show its sufficiency, let V,,(A) be a unitary U,(m)@ U,(n) module and let (J be the naturally induced invariant sesquilinear form of type B=O on V(A). It then suffices to show that (J is positive definite. We proceed by induction on the different Z-graded levels. Now (J is positive definite on V,(A) by definition. Assume that it is also positive definite on V,- *(A), k> 1. Then for any Ofu E V,(A) belonging to an irreducible U,(m)@ U,(n) module of highest weight ED+, Eq. (29) holds. Since Rviu E V,-,(h) the inductive hypothesis implies that the rhs is non- negative. In fact, it cannot vanish otherwise we would have


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which would imply

O=~ q’2’,~“(R,i)tR,iU=77U=[~*(C)-~V(C)]V i=l

in contradiction to Eq. (28). Thus the rhs of (29) must be strictly positive which implies, in view of Eq. (28), that (v,v)>O. This completes the first part of the argument.

As to the general case, we observe that every nonzero u E V,(h) is expressible as a sum

u=c u*, u,+o, (I where each u, belongs to a (possible multiple of a) definite irreducible U,(m) @ U,(n) module. Hence we obtain, in view of the first step,

(v,v,=C (v,,v,)>O,

a

where we have used the fact that U,(m)@ U,(n) submodules with different highest weights are orthogonal under the induced form. This proves (u,u)>O for all nonzero u E V,(A), from which the result follows by induction. 0

Our aim now is to reformulate the above result to obtain a necessary and sufficient condition for type (1) unitarity in terms of highest weights. In view of Proposition 1, this also implicitly classifies the unitary irreps of type (2).

VI. CLASSIFICATION OF UNITARY IRREPS OF U,(m[n)

Here, we classify those irreducible UJmln) modules V(A), A ED, , which are unitary for all values of q>O. In particular, such modules must also be unitary for gl( m/n) in the classical limit q-+1. Since the latter have been classified18%‘9 it remains now to determine which of these irreps give rise to unitary irreps of U,(mln). We have already noted that the defining irrep with highest weight A= e1 is undeformed and gives rise to a type (1) unitary irrep of UJ m In). Thus the dual irrep with highest weight A=-S,, is unitary of type (2).

It follows that all h-reps obtained from repeated tensor products of the vector irrep mTTE1 are unitary of type (1): These comprise the so-called contravariant tensor irreps.5*34335 Thus their duals, comprising the so-called covariant tensor irreps,34735 must be unitary of type (2). Thus we immediately arrive at:

Proposition 4: The contravariant [resp. covariant] tensor irreps are unitary irreps of UJmln) of We (1) [rev. C31. 0

For gl(mln) there exists, in addition to these tensor irreps, a much larger class of nontensorial typical unitary h-reps. We thus find it convenient, as in the classical case,” to consider the typical and atypical unitary irreps separately.

The highest weights of the atypical unitary irreps of gl(mln) have been classified in Ref. 19 (see Theorems 3 and 4) and they are all tensor irreps (or their tensor products with a real one- dimensional irrep). It follows from Proposition (4), that all such irreps are also unitary for the quantum case. Thus we immediately arrive at (Ref. 19)

Theorem 1: Suppose A ED + is atypical. Then


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(i) V(A) is unitary of type (I) if and only if A is real and there exists an odd index pE{1,2,...,n} such that

(A+p,e,- ~,>=(A,Ls,- s,,)=o.

(ii) V(A) is unitary of type (2)’ if and only if A is real and there exists an even index iE{1,2 ,...,m} such that

Note: The lowest weights of the above unitary irreps have been determined explicitly in Re$ 19. 0

It remains to investigate the type (1) and (2) typical unitary irreps, which require a bit more work. We concentrate first on the type (1) case. In Ref. 19, it was shown that a typical gl(mln) module V(A), A ED + , is unitary of type (1) if and only if A is real and

(A+p,e,- S,J>O. (30)

Thus Eq. (30) is certainly a necessary condition in order for a typical UJmln) module V(A) be type (1) unitary, for all q >O. Below, we show that this condition is also sufficient.

We require first a technical result on the weight spectrum of the irreducible modules V(A), with A satisfying Eq. (30). Following Ref. 19, we have:

Lemma (2): Suppose A ED, satisfies Eq. (30). Then

(A+p,PPO, VP=@+ :!) (v,/?)>O, V@E@,:, &I(A). cl

Following the notation of Ref. 19, if f3cQ:, we set

P,(e)=; Ix p; fi=e

in the case 13 is the empty set we interpret this as the zero weight. We note that the highest weights of the irreducible U,(m) @ U,(n) modules contained in a finite dimensional irreducible Us( mln) module V(A) are all of the form (Refs. 25 and 19):

v=A-2p,(B) (31)

for some eC@:. Now let C be the UJmln) Casimir invariant considered earlier; on an irreducible UJmln) module V(A) the eigenvalue of this invariant is given by Eq. (26). We have:

Proposition 5: Suppose A ED + is real and satisjes Eq. (30). If q > 1, then

with equality if and only if .!I is the empty set. Proof: We proceed in two steps. Step a: First, we assume 0 is a set of the following special form

e={Ei-Sp ,,..., Ei-Spt}, 1~~l<f.k2<...<~k<n. (9

Then, from Eq. (26), we have


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M. D. Gould and M. Scheunert: Unitary irreps for U,[gl(mjn)J 449

n

+c [4 2(A+P-2P,(e),s,)_q2(A+~,~~)]

p=l

= q2CA+P.‘i)(l-q-2k)+~ q2(A+P,Jp>(q-2- 1)

1=1

= 4 2(A+P.%)(l -q-2k)+q2(*+p.8& q2(~+PJ&-$&p~ 1).

l=l

Under the assumption q> 1, the second term above is negative and since (A + p, CT,+ - ~7~~) S (p,S,, - SC(,) = pl - &, forl<k,wehave

Xh(C)-XA-2p,(o)WP~ I(*+P..r)(l-q-2’)+q2(*+P,61,~);: q2(~r~e)(q-2-1). l=l

Now, observe that for l<k, p,k-,uILk-l, SO that

k xA(C)-~A-2p,(e~(C)~~2(A+p~ci)(l -q-2k)+q2(A+p*6Pk)C q2(lmk)(qm2- 1)

1= 1

= 4 2(A+p+Ci)( 1 -4-2k) +q2(A+p,8Fk) * -q-2k

=(1-q-=)[q 2(A+P,8i)-q2(A+P.6~~)]

=(I -q-2k)q2(*+P,6,t)(q2(*+P,Ei-~~~)- 1).

From Lemma 2, (A + p, Ei - SC(,) > 0 and since q> 1, we thus arrive at

with equality if and only if k=O and 6 is the empty set. This proves the result for 0 a set of the special form (*).

Step b: We now observe that every &@: can be written as a disjoint union

e= L Bi, i=l

with ei a set of the form (*). For each odd index ,u we similarly have the sets

so that 0 may be alternatively partitioned

Obviously, we have


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plw=5 P1(4)= Sl dq. i=l fl=l

Now from Step a above, we have, provided 0, is not the empty set,

> 5 [q2(*+P,~~)-q2(*+P-2Pl(~i),~~)I I*=1

q2(A+p.SJ[1 -q-2(2~,($).$)]

/L=l

W+P.$,(~ -s-2). /.&=I

Ei--ShEBi In view of this inequality, we thus obtain, provided 0 is not the empty set,

2(*+~-2p,(~),S~)_~2(A+p.6~)1

p=l

m ei-6GEei >C C 42(~+~J,J(l-q-2)+ 5 q2(*+~,s~)[q-2(2~1(e).~~)-1]

i= 1 P /L=l

= 5 q2(*+P~~p)1~pI( 1 -q-2)+ 2 q2(*+pJp)(q-21epl~ 1), p=l /L=l

where 1 epl is the cardinality of the set op. Thus,

. 1 Since, for (e,pi,

-2ie,i- 1

by- qq-2-l lq- 1 IS&J- 1

=Ie,I- C qe21= C (1-qe2’)20, for q>l I=0 I=0

we finally arrive at

for 19 not equal to the empty set. This is sufficient to prove the result. 0 Since the highest weights VE D + of the irreducible UJ m) @ UJ n) modules contained in the

irreducible module V(A) are all of the form (31), as noted previously, it follows by combining the results of Propositions 3 and 5 that: every irreducible module V(A), with A real and satisfying Eq.


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M. D. Gould and M. Scheunert: Unitary irreps for u,[gl(mln)] 451

(30), is a type (I) unitary module provided q > 1. In fact, we can assume the result for q z 1 since it has already been established for the classical case q--+1 (Refs. 18 and 19). Moreover, since the defining relations for UJ m In) are symmetric under the interchange q +-+q -‘, this latter result must also hold for q-‘a 1 (i.e., O<qCl). [This can in fact be established directly by following a similar procedure to the one above, but working with the Casimir invariant arising from the dual of the defining irrep.]

Thus we have shown, for A ED + real, that Eq. (30) is both necessary and sufficient in order for an irreducible typical module V(A) be type (1) unitary (for all q>O). Their corresponding duals, the highest weights of which are given in Ref. 19, thus comprise the typical irreducible unitary modules of type (2). We thereby arrive at (Ref. 19):

Theorem 2: Suppose he D + is real and typical. Then

(i) V(A) is unitary of type (I) if and only if

(A+tp,~,- S,)>O.

(ii) V(A) is unitary of type (2) if and only if

VII. CONCLUSIONS

We have presented a systematic classification of the finite dimensional unitary irreps of U,[gl(mln)] for generic q>O. Our main results are Theorems 1 and 2, which give a complete classification in terms of highest weights. It is shown that a given irreducible U,[gl(m In)] module V(A), with A ED + real, is type (1) unitary for all q>O iff

(A+P,E~- S,)>O; or ii:) th ere exists an odd index pE{1,2,...,n} such that

Similarly V(A) is type (2) unitary for all q>O iff

ii!) (A+P,E,-8,)<0; or there exists an even index iE{1,2,...,m} such that

These results demonstrate that all unitary irreps for the LSA gl(mln) quantize to give unitary irreps of the corresponding quantum supergroup. As noted in the Introduction, such irreps are most likely to be of physical interest where unitarity is a basic requirement. It would therefore be of interest to extend these results to the type I quantum supergroup U4[ C( n)] .

It is worth noting, for arbitrary real A ED + , that

A,-A+LY~ ~ifas SP i=l /&=l

is typical and V(A,) is unitary for real cy>O sufficiently large. In this way we obtain one-parameter families of (nonequivalent) typical unitary irreps. In Ref. 23 a particular family of such irreps was investigated for U,[g1(211)] and applied to obtain a new two variable link polynomial which coincided with that due to Kauffman36 but only for a particular choice of the parameter (Y. It is suggestive that such two variable link polynomials can be constructed corresponding to each of the


On: Tue, 18 Oct 2016 05:50:29


above one-parameter families. A systematic investigation of these link polynomials would therefore be of interest and may well shed new light on the construction and classification of two variable link polynomials.

Finally, it should be emphasized that in the above we have classified the class of irreps of U,(mln) which are unitary for all q>O. However, it is conceivable that additional irreps may exist which are unitary for specific values of the parameter q # 1 but nonunitary for y= 1. The following argument suggests though that this possibility cannot occur: under the assumption that the naturally induced forms (cf. Sec. III) on an irreducible module V(A), AED+, depend continuously on q>O the signature of the form must also depend continuously on q. Thus, if the form is positive definite for a particular value of q>O, it must be positive definite for ull q>O. Hence, if V(A) is a unitary U,(mln) module for a particular value of q>O, it must be unitary for all q>O.

ACKNOWLEDGMENTS

The authors take great pleasure in thanking J. R. Links, R. B. Zhang, and A. J. Bracken for many stimulating discussions. The present work was initiated during a visit of the second author to the Department of Mathematics of the University of Queensland. He gratefully acknowledges both the kind invitation and the hospitality extended to him.

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“M. Jimbo, Lett. Math. Phys. 10, 63 (1985); 11, 247 (1986). “P. P. Kulish and E. K. Sklyanin, in “Integrable Quantum Field Theories,” Lecture Notes in Physics 151, 61 (1982). ‘=R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic, London, 1982). “N. Yu. Reshetikhin, “Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links,” LOMI

preprints E-4-87, E-17-87. 14V. G. Turaev, Invent. Math. 92, 527 (1988). “R. B. Zhang, M. D. Gould, and A. J. Bracken, Commun. Math. Phys. 137, 13 (1991). 16M. Scheunen, W. Nahm, and V. Rittenberg, J. Math. Phys. 18, 146 (1977). “Y. Z. Zhang and M. D. Gould, J. Math. Phya. 34, 6045 (1993). “M. D. Gould and R. B. Zhang, J. Math. Phys. 31, 1524 (1990). 19M. D. Gould and R. B. Zhang, J. Math. Phys. 31, 2552 (1990). aOM Scheunert, M. D. Gould. and J. R. Links, to appear. “MI D. Gould, Lett. Math. Phys. 24, 183 (1992). ‘=R. B. Zhang, M. D. Gould, and A. J. Bracken, Nucl. Phys. B 354, 625 (1991). “5. R. Links and M. D. Gould, Lett. Math. Phys. 26. 187 (1992). 24M. D. Gould, R. B. Zhang, and A. J. Bracken, Bull. Austral. Math. Sot. 47, 353 (1993). 25 V. G. Kac, “Lecture Notes in Mathematics,” Vol. 676 (Springer, Berlin, 1978), p. 597. 26S. M. Khoroshkin and V. N. Tolstoy, Commun. Math. Phys. 141, 599 (1991). *‘S M. Khoroshkin and V. N. Tolstoy, Lett. Math. Phys. 24. 231 (1992). ‘sd. S. McAnally and M. D. Gould (to appear). 29R. B. Zhang, J. Math. Phys. 34, 1236 (1993). 30R. B. Zhang and M. D. Gould, J. Math. Phys. 32, 3261 (1991). 3’ M. Scheunert, “Lecture Notes in Math,” Vol. 716 (Springer, Berlin, 1979). 32M. D. Gould, J. R. Links, and A. J. Bracken, I. Math. Phys. 33, 1008 (1992). 33M. D. Gould, R. B. Zhang, and A. J. Bracken, J. Math. Phys. 32, 2298 (1991). 34P. H. Dondi and P. D. Jarvis, J. Phys. A 14, 547 (1981). 35A. B. Balantekin and I. Bars. J. Math. Phys. 22, 1149 (1981). 36L. H. Kauffman, Trans. Am. Math. Sot. 318, 417 (1990).


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