Induced representations of the two parametric quantum deformation U pq [ gl (2/2)]Nguyen Anh Ky Citation: Journal of Mathematical Physics 41, 6487 (2000); doi: 10.1063/1.1286510 View online: http://dx.doi.org/10.1063/1.1286510 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/41/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Approach of spherical harmonics to the representation of the deformed su(1,1) algebra J. Math. Phys. 49, 113511 (2008); 10.1063/1.3025922 Structure and representations on the quantum supergroup OSP q (2|2n) J. Math. Phys. 41, 6639 (2000); 10.1063/1.1286881 U q [sl (2| 1)] vertex operators, screen currents, and correlation functions at an arbitrary level J. Math. Phys. 41, 5277 (2000); 10.1063/1.533409 Master function approach to quantum solvable models on SL (2,c) and SL (2,c)/ GL (1,c) manifolds J. Math. Phys. 41, 505 (2000); 10.1063/1.533150 Two-parameter nonstandard deformation of 2×2 matrices J. Math. Phys. 40, 3553 (1999); 10.1063/1.532907

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Induced representations of the two parametric quantumdeformation U pq †gl „2Õ2…‡

Nguyen Anh KyDepartment of Physics, Chuo University, Kasuga, Bunkyo-ku Tokyo 112-8551, Japan andInstitute of Physics, P.O. Box 429, Bo Ho, Hanoi 10000, Vietnam

~Received 31 January 2000; accepted for publication 6 March 2000!

The two-parametric quantum superalgebra Up,q@gl(2/2)# and its induced represen-tations are considered. A method for constructing all finite-dimensional irreduciblerepresentations of this quantum superalgebra is also described in detail. It turns outthat finite-dimensional representations of the two-parametric Up,q@gl(2/2)#, even atgeneric deformation parameters, are not simply trivial deformations from those ofthe classical superalgebra gl~2/2!, unlike the one-parametric cases. ©2000American Institute of Physics.@S0022-2488~00!04308-5#

LIST OF SYMBOLS

fidirmod~s!: finite-dimensional irreducible module~s!GZ basis: Gel’fand-Zetlin basisQGZ basis: quasi-Gel’fand-Zetlin basislin.env.$X%: linear envelope ofXp,q: the deformation parameters@x#[@x#p,q

5(qx2p2x)/q2p21: a pq-deformation of a number or an operatorx

Vlp,q

^ Vrp,q : a tensor product between two linear spacesVl

p,q and Vrp,q or a tensor

product between a Up,q@gl~2!l #-module Vlp,q and a Up,q@gl~2!r #-module

Vrp,q

Tp,q:V0p,q : a tensor product between two Up,q@gl~2!%gl~2!#-modulesTp,q andV0

p,q

@E, F%: supercommutator betweenE andF@E,F# r[EF2rFE: an r-deformed commutator betweenE andF

We hope the notations@x#[@x#p,q for quantum deformations,@m# for highest weights~signatures!in ~quasi-! GZ bases~m!, and@,# for commutators do not confuse the reader.

I. INTRODUCTION

Introduced in the 1980’s as a result of the study on quantum integrable systems and Yang–Baxter equations,1 the quantum groups2–7 have been intensively investigated in different aspects.Since then many~algebraic and geometric! structures and various applications of quantum~super!groups have been found~see in this context, for example, Refs. 8–11!. It turns out that quantumgroups are related to unrelated, at first sight, areas of both physics and mathematics~Refs. 8–15and references therein!. For applications of quantum groups, as in the non-deformed cases, weoften need their explicit representations. However, despite remarkable results in this direction, theproblem of investigating and constructing explicit representations of quantum groups, especiallythose for quantum superalgebras, is still far from being satisfactorily solved. Even in the case ofone-parametric quantum superalgebras, explicit representations are mainly known for quantum Liesuperalgebras of lower ranks and of particular types like Uq@osp(1/2)# and Uq@gl(1/n)# ~Refs.15–17!, while for higher rank quantum Lie superalgebras,18–23 besides someq-oscillator repre-sentations which are most popular among those constructed, we do not know so much about other

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 41, NUMBER 9 SEPTEMBER 2000

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representations, in particular, the finite-dimensional ones which in many cases are related totrigonometric solutions of the quantum Yang–Baxter equations.1,8 Some general aspects and themodule structure of finite-dimensional representations of the quantum superalgebras Uq@gl(m/n)#are considered in Ref. 21, but, unfortunately, their explicit construction is still absent. So farexplicit finite-dimensional irreducible representations are all known and classified only for thoseUq@gl(m/n)# with both m andn<2 ~see Refs. 15, 22, and 23!.

What about multi-parametric deformations~first considered in Ref. 4!? This area is even lesscovered and results are much poorer. Some kinds of two-parametric deformations have beenconsidered by several authors from different points of view~see Refs. 24 and 25 and referencestherein! but, to our knowledge, explicit representations are known and/or classified in a few lowerrank cases such as Up,q@sl(2/1)# and Up,q@gl(2/1)# only.25,26 The latter two-parametric quantumsuperalgebra Up,q@gl(2/1)# was consistently defined and investigated in Ref. 25 where all itsfinite-dimensional irreducible representations were explicitly constructed and classified at genericdeformation parameters. This Up,q@gl(2/1)#, however, is still a small quantum superalgebra whichcan be defined without the so-called extra-Serre defining relations27–29 representing additionalconstraints on odd Chevalley generators in higher rank cases. Now, in order to include the extra-Serre relations on examination we consider a bigger two-parametric quantum superalgebra,namely Up,q@gl(2/2)#, and its representations. This quantum superalgebra Up,q@gl(2/2)# resemblesto the one-parametric quantum superalgebra UApq@gl(2/2)# but cannot be identified with the latter.Here we supposepÞq, otherwise we should return to the case of Uq@gl(2/2)# investigated alreadyin Refs. 22 and 23. Another motivation is that already in the non-deformed case, the superalgebrasgl(n/n), especially, their subalgebras sl(n/n) and psl(n/n), have special properties@in compari-son with other gl(m/n), mÞn# and, therefore, attract interest.30 Additionally, structures of two-parameter deformations considered in Ref. 25 and here are, of course, richer than those of one-parameter deformations. Every deformation parameter can be independently chosen to take aseparate generic value~including zero! or to be a root of unity.

Combining the advantages of the previously developed methods for Uq@gl(2/2)# andUp,q@gl(1/2)# ~see Refs. 22, 23, and 25! we can construct all finite-dimensional representations ofthe two-parametric quantum Lie superalgebra Up,q@gl(2/2)#. In the framework of this article weconsider representations at genericp and q only ~i.e., p and q are not roots of unity!, whilerepresentations at roots of unity are a subject of later separate investigations. In comparison withprevious papers,22,25 the approach here is somewhat modified because of some specific featuresarising in the present case but the main steps in the construction procedure remain the same.Following this approach we can directly construct explicit representations of the quantum super-algebra Up,q@gl(2/2)# induced from some~usually, irreducible! finite-dimensional representationsof the even subalgebra Up,q@gl~2!% gl~2!#, which itself is a quantum algebra. Since the latter is astability subalgebra of Up,q@gl(2/2)# we expect the representations of Up,q@gl(2/2)# constructedare decomposed into finite-dimensional irreducible representations of Up,q@gl~2!% gl~2!#. For aclear description of this decomposition we shall introduce a Up,q@gl(2/2)#-basis ~i.e., a basiswithin a Up,q@gl(2/2)#-module or briefly a basis of Up,q@gl(2/2)# which will be convenient for usin investigating the module structure. This basis@see~4.26!# can be expressed in terms of somebasis of the even subalgebra Up,q@gl~2!% gl~2!# which in turn represents a~tensor! product be-tween two Up,q@gl~2!#-bases referred to as the left and the right ones. As is shown in Ref. 25, theGel’fand–Zetlin~GZ! patterns can serve again as a basis of finite-dimensional representations ofUp,q@gl~2!#. Thus, finite-dimensional representations of Up,q@gl~2!% gl~2!# are realized in tensorproducts of two such Up,q@gl~2!# GZ bases. For genericp and q, the finite-dimensionalUp,q@gl(2/2)#-modules constructed have similar structures to those of Uq@gl(2/2)# investigated inRefs. 22 and 23 and to those of gl~2/2! investigated in Ref. 31. However, finite-dimensionalrepresentations of Up,q@gl(2/2)# at generic deformation parameters are not simply trivial defor-mations from those of gl~2/2!, that is, the former cannot be obtained from the latter by puttingquantum deformation brackets in appropriate places, unlike many cases of one-parametric defor-mations. When one or both ofp andq are roots of unity the structures of Up,q@gl(2/2)#-modulesare drastically different, but we hope that the present method for construction of finite-dimensional

6488 J. Math. Phys., Vol. 41, No. 9, September 2000 Nguyen Anh Ky

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representations of Up,q@gl(2/2)# at generic deformation parameters can be extended on its finite-dimensional representations at roots of unity.

This paper is organized as follows. The two-parametric quantum superalgebra Up,q@gl(2/2)# isconsistently defined in Sec. II where we also describe how to construct its representations inducedfrom representations of the stability subalgebra Up,q@gl~2!% gl~2!#. Section III is devoted to con-structing finite-dimensional representations of Up,q@gl(2/2)#. Finally, some comments and con-clusions are made in Sec. IV.

II. THE QUANTUM SUPERALGEBRA U p,q†gl „2Õ2…‡

The quantum superalgebra Up,q[Up,q@gl(2/2)# as a two-parametric deformation of the uni-versal enveloping algebra U@gl~2/2!# of the Lie superalgebra gl~2/2! is generated by the operatorsLk , E12, E23, E34, E21, E32, E43, andEii (1< i<4) called again Cartan–Chevalley generatorsand satisfying32 the following.32

~a! The super-commutation relations~1< i , i 11,j , j 11<4! are

@Eii ,Ej j #50, ~2.1a!

@Eii ,Ej , j 11#5~d i j 2d i , j 11!Ej , j 11 , ~2.1b!

@Eii ,Ej 11,j #5~d i , j 112d i j !Ej 11,j , ~2.1c!

@even generator,Lk#50, k51,2,3, ~2.1d!

@Ei ,i 11 ,Ej 11,j%5d i j S q

pD Li2hi ~11d i2!/2

@hi #, ~2.1e!

with hi5@Eii 2(di 11 /di)Ei 11,i 11#, L15Ll , L250, L35Lr , andd15d252d352d451.~b! The Serre relations are

@E12,E34#5@E21,E43#50, ~2.2a!

E232 5E32

2 50, ~2.2b!

@E12,E13#p5@E21,E31#q5@E24,E34#q5@E42,E43#p50. ~2.2c!

~c! The extra-Serre relations are

$E13,E24%50, ~2.3a!

$E31,E42%50. ~2.3b!

Here, the operators

E13ª@E12,E23#q21, ~2.4a!

E24ª@E23,E34#p21, ~2.4b!

E31ª2@E21,E32#p21, ~2.4c!

E42ª2@E32,E43#q21, ~2.4d!

and the operators composed in the following way,

E14ª@E12,@E23,E34#p21#q21[@E12,E24#q21, ~2.5a!

6489J. Math. Phys., Vol. 41, No. 9, September 2000 Induced representations of the two-parametric . . .

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E41ª@E21,@E32,E43#q21#p21[2@E21,E42#p21, ~2.5b!

are defined as new generators which, likeE23 andE32, are all odd and have vanishing squares.These generatorsEi j , 1< i , j <4, are two-parametric deformation analogs~pq-analogs! of theWeyl generatorsei j , 1< i , j <4, of the superalgebra gl~2/2! whose universal enveloping algebraU@gl~2/2!# is a classical limit of Up,q@gl(2/2)# whenp,q→1. The so-called maximal-spin opera-tors Lk are constants within a Up,q@gl(2)#-fidirmod and are different for differentUp,q@gl(2)#-fidirmods. Therefore, commutators between these operators with the odd generatorsintertwining Up,q@gl(2)#-fidirmods take concrete forms on concrete basis vectors. Other commu-tation relations betweenEi j follow from the relations~2.1!–~2.3! and the definitions~2.4! and~2.5!.

The subalgebra Up,q@gl(2/2)0#(,Up,q@gl(2/2)#0,Up,q@gl(2/2)#) is even and isomorphic toUp,q@gl(2)% gl(2)#[Up,q@gl(2)# % Up,q@gl(2)#, which is completely defined byL1 , L3 , E12,E34, E21, E43, andEii , 1< i<4,

Up,q@gl~2/2!0#5 lin. env.$L1 ,L3 ,Ei j i i , j 51,2 and i , j 53,4%. ~2.6!

In order to distinguish two components Up,q@gl(2)# of Up,q@gl(2/2)0# we set

left Up,q@gl~2!#[Up,q@gl~2! l #ª lin. env.$L1 ,Ei j i i , j 51,2%, ~2.7!

right Up,q@gl~2!#[Up,q@gl~2!r #ª lin. env.$L3 ,Ei j i i , j 53,4%, ~2.8!

that is,

Up,q@gl~2/2!0#5Up,q@gl~2! l % gl~2!r #. ~2.9!

Looking at the relations~2.1!–~2.3! we see that the odd spacesA1 andA2 spanned on thepositive and negative odd roots~generators! Ei j andEji , 1< i<2, j <4, respectively,

A15 lin. env.$E14,E13,E24,E23%, ~2.10!

A25 lin. env.$E41,E31,E42,E32%, ~2.11!

are representation spaces of the even subalgebra Up,q@gl(2/2)0# which, as seen from~2.1! and~2.2!, is a stability subalgebra of Up,q@gl(2/2)#. Therefore, we can construct representations ofUp,q@gl(2/2)# induced from some~finite-dimensional irreducible, for example! representations ofUp,q@gl(2/2)0# which are realized in some representation spaces~modules! V0

p,q representingtensor products of Up,q@gl(2)l #-modulesV0,l

p,q and Up,q@gl(2)r #-modulesV0,rp,q :

V0p,q~L!5V0,l

p,q~L l ! ^ V0,rp,q~L r !, ~2.12!

whereL’s are some signatures~such as highest weights, respectively! characterizing the modules~highest weight modules, respectively!. HereL l and L r are referred to as the left and the rightcomponents ofL, respectively,

L5@L l ,L r #. ~2.13!

If we demand

E23V0p,q~L!50, ~2.14!

hence

Up,q~A1!V0p,q50, ~2.15!

6490 J. Math. Phys., Vol. 41, No. 9, September 2000 Nguyen Anh Ky

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we turn the Up,q@gl(2/2)0#-moduleV0p,q into a Up,q(B)-module where

B5A1 % gl~2! % gl~2!. ~2.16!

The Up,q@gl(2/2)#-moduleWp,q induced from the Up,q@gl(2/2)0#-moduleV0p,q is the factor-space

Wp,q5Wp,q~L!5Up,q^ V0p,q~L!]/ I p,q~L!, ~2.17!

which, of course, depends onL, where

Up,q[Up,q@gl~2/2!#, ~2.18!

while I p,q is the subspace

I p,q5 lin. env.$ub^ v2u^ bviuPUp,q,bPUp,q~B!.Up,q ,vPV0p,q%. ~2.19!

Using the commutation relations~2.1!–~2.3! and the definitions~2.4! and ~2.5! we can prove thefollowing analog of the Poincare´–Birkhoff–Witt theorem.

Proposition 1: The quantum deformationUp,qªUp,q@gl(2/2)# is spanned on all possiblelinear combinations of the elements

g5~E23!h1~E24!

h2~E13!h3~E14!

h4~E41!u1~E31!

u2~E42!u3~E32!

u4g0 , ~2.20!

or equivalently

g5~E41!u1~E31!

u2~E42!u3~E32!

u4b, ~2.21!

where g0PUp,q@gl(2/2)0#, bPUp,q(B) and h i , u i50,1.Any vectorw from the moduleWp,q can be represented as

w5u^ v, uPUp,q , vPV0p,q . ~2.22!

ThenWp,q is a Up,q@gl(2/2)#-module in the sense

gw[g~u^ v !5gu^ vPWp,q, ~2.23!

for g, uPUp,q , wPWp,q, and vPV0p,q . Taking into account the fact thatV0

p,q(L) is aUp,q(B)-module we have

Wq~L!5 lin. env.$~E41!u1~E31!

u2~E42!u3~E32!

u4^ vivPV0p,q ,u1 ,...,u450,1%. ~2.24!

Consequently, a basis ofWp,q can be constituted by taking all the vectors of the form

uu1 ,u2 ,u3 ,u4 ;~l!&ª~E41!u1~E31!

u2~E42!u3~E32!

u4^ ~l!, u i50,1, ~2.25!

where~l! is a ~GZ, for example! basis ofV0p,q[V0

p,q(L). We refer to this basis ofWp,q as theinduced Up,q@gl(2/2)#-basis~or simply, the induced basis! in order to distinguish it from anotherUp,q@gl(2/2)#-basis introduced later and called a reduced basis, which is more convenient forinvestigating the module structure ofWp,q. It is obvious that if the moduleV0

p,q is finite dimen-sional so is the moduleWp,q. Finite-dimensional representations of Up,q@gl(2/2)# are namely thesubject of the next section.

6491J. Math. Phys., Vol. 41, No. 9, September 2000 Induced representations of the two-parametric . . .

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III. FINITE-DIMENSIONAL REPRESENTATIONS OF U p,q†gl „2Õ2…‡

In this section we consider finite-dimensional representations of Up,q@gl(2/2)# induced fromirreducible finite-dimensional representations of Up,q@gl(2/2)0#. We first construct the bases of themoduleWq and then find the explicit matrix elements for the finite-dimensional representations ofUp,q@gl(2/2)#.

We can show that the GZ patterns

Fm12 m22

m11G[F @m#

m11G , ~3.1!

wheremi j are complex numbers such thatm122m11PZ1 and m112m22PZ1 , can serve as abasis of a Up,q@gl(2)#-fidirmod. Indeed, finite-dimensional representations of Up,q@gl(2)# are highweight and if the operatorsL andEi j , i , j 51,2, are defined on the basis~3.1! as follows,

LFm12 m22

m11G5 1

2~ l 122 l 2221!Fm12 m22

m11G ,

E11Fm12 m22

m11G5~ l 1111!Fm12 m22

m11G ,

E22Fm12 m22

m11G5~ l 121 l 222 l 1112!Fm12 m22

m11G , ~3.2!

E12Fm12 m22

m11G5~@ l 122 l 11#@ l 112 l 22# !1/2Fm12 m22

m1111 G ,E21Fm12 m22

m11G5~@ l 122 l 1111#@ l 112 l 2221# !1/2Fm12 m22

m1121 G ,they really satisfy commutation relations of Up,q@gl(2)# given in ~2.1!. Here the notation

l i , j5mi , j2 i , i 51,2, ~3.3a!

and later also the notation

l i j8 5mi j 2 i 12, i 53,4, ~3.3b!

are used. Since the Up,q@gl(2/2)0#-fidirmod V0p,q is decomposed into a Up,q@gl(2)l #-fidirmod V0,l

p,q

and a Up,q@gl(2)r #-fidirmod V0,rp,q via the tensor product

V0p,q5V0,l

p,q^ V0,r

p,q , ~3.4!

its basis, therefore, is a tensor product

Fm13 m23

m11G ^ Fm33 m43

m31G[F @m# l

m11G ^ F @m# r

m31G[~m! l ^ ~m!r[~m! ~3.5!

between a GZ basis ofV0,lp,q spanned on the vectors (m) l and a GZ basis ofV0,r

p,q spanned on thevectors (m) r . Following the approach of Ref. 22~see also Refs. 23 and 31! and keeping thenotations used there, we can represent the basis~3.5! of V0

p,q in the form

Fm13 m23

m11;m33 m43

m31G[F @m# l

m11;@m# r

m31G[~m!. ~3.6!

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Then, the signatureL, which now is the highest weight, is given by the first row@m13,m23,m33,m43#[@@m# l ,@m# r #[@m# common for all the basis vectors~3.6! of V0

p,q :

V0p,q[V0

p,q~L!5V0p,q~@m# !5Vo,l

p,q~@m# l ! ^ V0,rp,q~@m# r !. ~3.7!

The explicit action of Up,q@gl(2/2)0# on V0p,q(@m#) follows directly from ~3.2! and

g0~m!5g0,l~m! l ^ ~m!r1~m! ^ g0,r~m!r ~3.8!

for g0[g0,l % g0,rPUq@gl(2/2)0# and (m)PV0q(@m#).

The basis vector withm115m13 andm315m33,

Fm13 m23

m13;m33 m43

m33G[F @m# l

m13;@m# r

m33G[~M !, ~3.9!

satisfying the conditions

Eii ~M !5mi3~M !, i 51,2,3,4,~3.10!

E12~M !5E34~M !50,

is the highest-weight vector inV0p,q(@m#). Therefore, as in the classical case (p5q51)31 and in

the case of one-parametric deformation (p5q),22 the highest-weight@m# is nothing but an orderedset of the eigenvalues of the Cartan generatorsEii on the highest-weight vector~M!. The latter isalso the highest-weight vector inWp,q(@m#) because of the condition~2.14!. All other, i.e., lowerweight, basis vector ofV0

p,q can be obtained from the highest-weight vector~M! through acting onthe latter by monomials of the lowering generatorsE21 andE43 in definite powers:

~m!5S @m112m23#! @m312m43#!

@m132m23#! @m132m11#! @m332m43#! @m332m31#!D 1/2

~E21!m132m11~E43!

m332m31~M !,

~3.11!

where@n#’s stand for

qn2p2n

q2p21 [@n#p,q[@n#, ~3.12!

while

@n#! 5@1#@2#¯@n21#@n#. ~3.13!

Therefore, the induced basis~2.25! of Wp,q(L)5Wp,q(@m#) now takes the form

uu1u2 ,u3 ,u4 ;~m!&ª~E41!u1~E31!

u2~E42!u3~E32!

u4^ ~m!. ~3.14!

The subspaceTp,q consisting of

uu1 ,u2 ,u3 ,u4&ª~E41!u1~E31!

u2~E42!u3~E32!

u4 ~3.15!

can be considered as a Up,q@gl(2/2)0#-adjoint module which is 16-dimensional when all theu i

( i 51,2,3,4) take all two possible values 0 and 1, that is( i 514 u i runs all over the range from 0 to

4. ThusWp,q(@m#) being a tensor product between two Up,q@gl(2/2)0#-modules,

Wp,q~@m# !5Tp,q(V0p,q~@m# !, ~2.248!

is, in general, a reducible Up,q@gl(2/2)0#-module and is decomposed into irreducibleUp,q@gl(2/2)0#-submodules. We arrive at the next assertion.

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Proposition 2: The inducedUp,q@gl(2/2)#-module Wp,q is the linear span

Wp,q~@m# !5 lin. env.$~E41!u1~E31!

u2~E42!u3~E32!

u4^ vivPV0p,q~@m# !,u i50,1%,

~2.2488!

which is decomposed into a direct sum of (16, at most)Up,q@gl(2/2)0#-fidirmods Vkp,q(@m#k):

Wp,q~@m# !5 %

k50

15

Vkp,q~@m#k!, ~3.16!

where@m#k are signatures of Vkp,q[Vk

p,q(@m#k).Here, we call@m#k[@m12,m22,m32,m42#k the local highest weights of the submodulesVk

p,q intheir GZ bases denoted now as

Fm12 m22

m11;M32 m42

m31G

k

[~m!k . ~3.17!

The highest-weight@m#0[@m# of V0p,q being also the highest weight ofWp,q is referred to as the

global highest weight. We call@m#k , kÞ0, the local highest weights in the sense that theycharacterize the submodulesVk

p,q,Wp,q as Up,q@gl(2/2)0#-fidirmods only, while the global high-est weight@m# characterizes the Up,q@gl(2/2)#-moduleWp,q as the whole. In the same way wedefine the local highest-weight vectors (M )k in Vk

p,q as those (m)k satisfying the conditions@cf.~3.10!#

Eii ~M !k5mi2~M !k , i 51,2,3,4,~3.18!

E12~M !k5E34~M !k50.

The highest-weight vector~M! of V0p,q is also the global highest-weight vector inWp,q for which

the condition@see~2.14!#

E23~M !50 ~3.19!

and the conditions~3.18! simultaneously hold.Let us denote byGk

p,q the basis system spanned on the basis vectors (m)k in ~3.17! in eachVk

p,q(@m#). For a basis ofWp,q we can choose the unionGp,q5øk5015 Gk

p,q of all the basesGkp,q ,

namely, a basis vector ofWp,q has to be identified with one of the vectors (m)k , 0<k<15. ThebasisGp,q is referred to as the Up,q@gl(2/2)#-reduced basis or, simply, the reduced basis. It is clearthat every basisGk

p,q5Gk(@m#k)p,q is labeled by a local highest-weight@m#k , while the basis

Gp,q5Gp,q(@m#) is labeled by the global highest-weight@m#. Going ahead, we modify the nota-tion ~3.17! for the basis vectors inGp,q as follows@cf. ~4.26! in Ref. 22#

Fm13 m23 m33 m43

m12 m22 m32 m42

m11 0 m31 0G

k

[Fm12 m22

m11;m32 m42

m31G

k

[~m!k , ~3.20!

with k running from 0 to 15 as fork50 we must take into accountmi25mi3 , i 51,2,3,4, i.e.,

~m!0[~m!5Fm13 m23 m33 m43

m13 m23 m33 m43

m11 0 m31 0G . ~3.21!

In ~3.20! the first row @m#5@m13,m23,m33,m43# being the~global! highest-weight ofWp,q isfixed for all the vectors in the wholeWp,q and characterizes this module itself, while the second

6494 J. Math. Phys., Vol. 41, No. 9, September 2000 Nguyen Anh Ky

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row is a ~local! highest weight of some submoduleVkp,q and tells us that the considered basis

vector (m)k of Wp,q belongs to this submodule in the decomposition~3.16! corresponding to thebranching rule Up,q@gl(2/2)#.Up,q@gl(2/2)0#.Up,q@gl~1!^gl(1)#. We refer to ~3.20! as thequasi-Gel’fand–Zetlin~QGZ! basis.

It is easy to see that the highest-weight vectors (M )k in the notation~3.20! are

~M !k5F m13 m23 m33 m43

m12 m22 m32 m42

m12 m22 m32 m42

m12 0 m32 0

Gk

, k50,1,...,15. ~3.22!

The ~global! highest-weight vector~M! in ~3.9! is given now by

~M !5Fm13 m23 m33 m43

m13 m23 m33 m43

m13 0 m33 0G . ~3.23!

A highest-weight vector (M )k expressed in terms of the induced basis~3.14! has the form of ahomogeneous polynomial of a definite degreeh in negative odd generators (Ei j ,1< j <2, i<4)acting on (m)PV0

p,q(@m#):

~M !k[~M !h,h5 (u i50,1

Ch,h~u1 ,u2 ,u3 ,u4!uu1 ,u2 ,u3 ,u4 ;~m!& ~3.24!

with h5( i 514 u i fixed for every (M )h,h , and the coefficientsCh,h determined by solving Eqs.

~3.18!. Applying ~3.11! to any (M )h,h we find all the basis vectors (m)h,h of the correspondingfidirmod Vh,h

p,q , which is a linear space spanned on homogeneous polynomials of the negative oddgenerators of the same degreeh since~3.11! does not changeh. Here we callh the level ofVh,h

p,q .It is easy to see that on the levelh50 there is only one fidirmod, namelyV0

p,q[V0,1p,q , while on the

next levelh51 there are four fidirmods, say,V1,hp,q , h51,2,3,4. On the levelh52 we can find six

fidirmodsV2,hp,q , 1<h<6, which are divided into two groups~h51 – 3 andh54 – 6! expressed in

terms of two independent groups of second-order monomials of odd generators~3.15! acting on~m!. For h53 the number of fidirmods is four,V3,h

p,q , h51,2,3,4, and finally onh54 we findagain only one fidirmodV4,1

p,q . However, this form~3.24!, which was used in the one-parametriccase,22 is now inconvenient for us here to apply formula~3.11! in order to find all other~i.e., lowerweight! reduced basis vectors. It is so because of the presence of the maximal-spin operatorsLi

which are not diagonalized in the induced basis but in the reduced basis@since an eigenvalue ofanyLi is a fixed constant only within a Up,q@gl(2)#-fidirmod ~or fidirmod, for short! and changesfrom fidirmod to fidirmod#. Applying ~3.11! we have to push generatorsE21 andE43 to the rightside until reachingV0

p,q by using commutation relations~2.1! and ~2.2! which give rise toLi ’sacting on the induced basis vectors. But it is extremtly difficult to get explicit actions ofLi on thelatter vectors before knowing how they are projected on the reduced basis which, however, we arenow looking for. Instead, we will write down (M )k[(M )h,h in a form convenient for applying~3.11! which leaves theh’s unchanged:

~M !0[~M !0,15a0u0,0,0,0;~M !&[~M !, a0[1,

~M !1[~M !1,15a1u0,0,0,1;~M !&[a1[E32~M !,

~M !2[~M !1,25a2H 1

a1E21~M !12

@2l 11#

@2l #E32E21~M !J ,

6495J. Math. Phys., Vol. 41, No. 9, September 2000 Induced representations of the two-parametric . . .

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~M !3[~M !1,35a3H 1

a1E43~M !12

@2l 811#

@2l 8#E32E43~M !J ,

~M !4[~M !1,4

5a4H 1

a1E21E43~M !12

1

a2E43~M !22

1

a3E21~M !32

@2l 11#@2l 811#

@2l #@2l 8#E32E21E43~M !J ,

~M !5[~M !2,15a5u0,0,1,1;~M !&[a5E42E32~M !,

~M !6[~M !2,25a6H 1

a5E21~M !52

@2l 12#

@2l #E42E32E21~M !J ,

~M !7[~M !2,35a7H 1

a5E21

2 ~M !521

a6

@2#@2l 11#

@2l #E21~M !62

@2l 11#@2l 12#

@2l #@2l 21#E42E32E21

2 ~M !J ,

~3.25!~M !8[~M !2,45a8u0,1,0,1;~M !&[a8E31E32~M !,

~M !9[~M !2,55a9H 1

a8E43~M !82

@2l 812#

@2l 8#E31E32E43~M !J ,

~M !10[~M !2,6

5a10H 1

a8E43

2 ~M !821

a9

@2#@2l 811#

@2l 8#E43~M !92

@2l 811#@2l 812#

@2l 8#@2l 821#E31E32E43

2 ~M !J ,

~M !11[~M !3,15a11u0,1,1,1;~M !&[a11E31E42E32~M !,

~M !12[~M !3,25a12H 1

a11E21~M !112

@2l 11#

@2l #E31E42E32E21~M !J ,

~M !13[~M !3,35a13H 1

a11E43~M !112

@2l 811#

@2l 8#E31E42E32E43~M !J ,

~M !14[~M !3,4

5a14H 1

a11E21E43~M !112

1

a12E43~M !122

1

a13E21~M !13

2@2l 11#@2l 811#

@2l #@2l 8#E31E42E32E21E43~M !J ,

~M !15[~M !4,15a15u1,1,1,1;~M !&[a15E41E31E42E32~M !,

where l 5 12 (m132m23) and l 85 1

2 (m332m43), while ak5ak(p,q) are coefficients depending, ingeneral, onp and q. Indeed, (M )k given in ~3.25! form a set of all linear-independent vectorssatisfying the conditions~3.18!. Looking at~3.25! we easily identify the highest weights@m#k :

@m#05@m13,m23,m33,m43#,

@m#15@m13,m2321,m3311,m43#,

@m#25@m1321,m23,m3311,m43#,

6496 J. Math. Phys., Vol. 41, No. 9, September 2000 Nguyen Anh Ky

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@m#35@m13,m2321,m33,m4311#,

@m#45@m1321,m23,m33,m4311#,

@m#55@m13,m2322,m3311,m4311,m4311#,

@m#65@m1321,m2321,m3311,m4311#6 ,

@m#75@m1322,m23,m3311,m4311#,~3.26!

@m#85@m1321,m2321,m3312,m43#,

@m#95@m1321,m2321,m3311,m4311#9 ,

@m#105@m1321,m2321,m33,m4312#,

@m#115@m1321,m2322,m3312,m4311#,

@m#125@m1322,m2321,m3312,m4311#,

@m#135@m1321,m2322,m3311,m4312#,

@m#145@m1322,m2321,m3311,m4312#,

@m#155@m1322,m2322,m3312,m4312#.

In the latest formula~3.26!, with the exception of@m#6 and@m#9 where a degeneration is present,we skip the subscriptk on the rhs. The proofs of~3.25! and ~3.26! follow from direct computa-tions.

Using the rule~3.11!, which now reads

~m!k5S @m112m22#! @m312m42#!

@m122m22#! @m122m11#! @m322m42#! @m322m31#!D 1/2

~E21!m122m11~E43!

m322m31~M !k ,

~3.118!

we can find all the basis vectors (m)k :

~m!05u0,0,0,0;~m!&,

~m!15a1qH 2S @ l 132 l 11#@ l 332 l 3111#

@2l 11#@2l 811# D 1/2

u1,0,0,0;~m!111&

2ql 82s8S @ l 132 l 11#@ l 312 l 4321#

@2l 11#@2l 811# D 1/2

u0,1,0,0;~m!111231&

1p2 l 1sS @ l 112 l 23#@ l 332 l 3111#

@2l 11#@2l 811# D 1/2

u0,0,1,0;~m!&

1p2 l 1sql 82s8S @ l 112 l 23#@ l 312 l 4321#

@2l 11#@2l 811# D 1/2

u0,0,0,1;~m!231&J ,

6497J. Math. Phys., Vol. 41, No. 9, September 2000 Induced representations of the two-parametric . . .

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~m!252a2qS q

pD l 2s21H S @ l 112 l 23#@ l 332 l 3111#

@2l #@2l 811# D 1/2

u1,0,0,0;~m!111&

1ql 82s8S @ l 112 l 23#@ l 312 l 4321#

@2l #@2l 811# D 1/2

u0,1,0,0;~m!111231&

1ql 1s11S @ l 132 l 11#@ l 332 l 3111#

@2l #@2l 811# D 1/2

u0,0,1,0;~m!&

1ql 1s1 l 82s811S @ l 132 l 11#@ l 312 l 4321#

@2l #@2l 811# D 1/2

u0,0,0,1;~m!231&J ,

~m!35a3H 2qS q

pD l 82s8S @ l 132 l 11#@ l 312 l 4321#

@2l 11#@2l 8# D 1/2

u1,0,0,0;~m!111&

1S q

pD 2l 8S @ l 132 l 11#@ l 332 l 3111#

@2l 11#@2l 8# D 1/2

u0,1,0,0;~m!111231&

1qp2 l 1sS q

pD l 82s8S @ l 112 l 23#@ l 312 l 4321#

@2l 11#@2l 8# D 1/2

u0,0,1,0;~m!&

2p2 l 1sS q

pD 2l 8S @ l 112 l 23#@ l 332 l 3111#

@2l 11#@2l 8# D 1/2

u0,0,0,1;~m!231&J ,

~m!45a4S q

pD l 2s1 l 82s821H qS @ l 112 l 23#@ l 312 l 4321#

@2l #@2l 8# D 1/2

u1,0,0,0;~m!111&

2p2 l 82s8S @ l 112 l 23#@ l 332 l 3111#

@2l #@2l 8# D 1/2

u0,1,0,0;~m!111231&

1ql 1s12S @ l 132 l 11#@ l 312 l 4321#

@2l #@2l 8# D 1/2

u0,0,1,0;~m!&

2ql 1s11p2 l 82s8S @ l 132 l 11#@ l 332 l 3111#

@2l #@2l 8# D 1/2

u0,0,0,1;~m!231&J ,

~m!55a5S q

pD l 82s811H S @ l 132 l 11#@ l 132 l 1121#

@2l 11#@2l 12# D 1/2

u1,1,0,0;~m!111111231&

1p2 l 1sS @ l 132 l 11#@ l 112 l 2311#

@2l 11#@2l 12# D 1/2

u0,1,1,0;~m!111231&

2p2 l 1s11S @ l 132 l 11#@ l 112 l 2311#

@2l 11#@2l 12# D 1/2

u1,0,0,1;~m!111231

1p2~2 l 1s!S @ l 112 l 23#@ l 112 l 2311#

@2l 11#@2l 12# D 1/2

u0,0,1,1;~m!231&J .

~m!65a6

a5S @2l 11#@2l 12#@ l 132 l 11#

@ l 112 l 2311# D 1/2

~m!52a6S q

pD l 82s811 @2l 12#

@2l #

3H @ l 132 l 1122#S @ l 132 l 1121#

@ l 112 l 2311# D1/2

u1,1,0,0;~m!111111231&

1p2 l 1s11@ l 132 l 1121#u0,1,1,0;~m!111231&2p2 l 1s12@ l 132 l 1121#u1,0,0,1;~m!111231&

6498 J. Math. Phys., Vol. 41, No. 9, September 2000 Nguyen Anh Ky

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1p22~ l 2s21!~ @ l 112 l 23#@ l 132 l 11# !1/2u0,0,1,1;~m!231&J ,

~m!75a7

a5S @2l 21#@2l #@2l 11#@2l 12#@ l 132 l 1121#@ l 132 l 11#

@ l 112 l 23#@ l 112 l 2311# D 1/2

~m!5

2a7

a6@2#@2l 11#S @2l 21#@ l 132 l 1121#

@2l #@ l 112 l 23#D 1/2

~m!6

2a7

@2l 11#@2l 12#

~@2l 21#@2l # !1/2 S q

pD l 82s811H @ l 132 l 1122#@ l 132 l 1123#

~@ l 112 l 23#@ l 112 l 2311# !1/2 u1,1,0,0;~m!111111231&

1p2 l 1s12@ l 132 l 1122#S @ l 132 l 1121#

@ l 112 l 23#D 1/2

u0,1,1,0;~m!111231&

2p2 l 1s13@ l 132 l 1122#S @ l 132 l 1121#

@ l 112 l 23#D 1/2

u1,0,0,1;~m!111231&

1p22~ l 2s22!~ @ l 132 l 1121#@ l 132 l 11# !1/2u0,0,1,1;~m!231&J ,

~3.27!

~m!85a8q2S q

pD l 2s21H S @ l 332 l 3111#@ l 332 l 3112#

@2l 811#@2l 812# D 1/2

u1,0,1,0;~m!111&

1q2l 8S @ l 332 l 3112#@ l 312 l 4321#

@2l 811#@2l 812# D 1/2

u1,0,0,1;~m!111231&

1q2l 811S @ l 332 l 3112#@ l 312 l 4321#

@2l 811#@2l 812# D 1/2

u0,1,1,0;~m!111231&

1q2~2l 811!S @ l 312 l 4322#@ l 312 l 4321#

@2l 811#@2l 812# D 1/2

u0,1,0,1;~m!111231231&J ,

~m!95a9

a8S @2l 811#@2l 812#@ l 332 l 3112#

@ l 312 l 4321# D 1/2

~m!82a9q2S q

pD l 2s-1 @2l 812#

@2l 8#

3H @ l 332 l 31#S @ l 332 l 3111#

@ l 312 l 4321# D1/2

u1,0,1,0;~m!111&1ql 82s821@ l 332 l 3111#

3u1,0,0,1;~m!111231&1ql 82s8@ l 332 l 3111#u0,1,1,0;~m!111231&

1q2~ l 82s8!~ @ l 332 l 3112#@ l 312 l 4322# !1/2u0,1,0,1;~m!111231231&J ,

~m!105a10

a8S @2l 821#@2l 8#@2l 811#@2l 812#@ l 332 l 3111#@ l 332 l 3112#

@ l 312 l 4322#@ l 312 l 4321# D 1/2

~m!8

2a10

a9@2#@2l 811#S @2l 821#@ l 332 l 3111#

@2l 8#@ l 312 l 4322# D 1/2

~m!92a10S q

pD l 2s21 @2l 811#@2l 812#

@2l 821#@2l 8#

3H q2@ l 332 l 3121#@ l 332 l 31#S @2l 821#@2l 8#

@ l 312 l 4322#@ l 312 l 4321# D1/2

u1,0,1,0;~m!111&

1ql 82s8@ l 332 l 31#S @2l 821#@2l 8#@ l 332 l 3111#

@ l 312 l 4322# D 1/2

u1,0,0,1;~m!111231&

6499J. Math. Phys., Vol. 41, No. 9, September 2000 Induced representations of the two-parametric . . .

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1ql 82s811@ l 332 l 31#S @2l 821#@2l 8#@ l 332 l 3111#

@ l 312 l 4322# D 1/2

u0,1,1,0;~m!111231&

1q2~ l 82s8!~ @2l 821#@2l 8#@ l 332 l 3111#@ l 332 l 3112# !1/2u0,1,0,1;~m!111231231&J ,

~m!115a11S q

pD l 2s1 l 82s811H pS @ l 132 l 1121#@ l 332 l 3112#

@2l 11#@2l 811# D 1/2

u1,1,1,0;~m!111111231&

1p2 l 1s12S @ l 112 l 2311#@ l 332 l 3112#

@2l 11#@2l 811# D 1/2

u1,0,1,1;~m!111231&

1ql 82s811pS @ l 132 l 1121#@ l 312 l 4322#

@2l 11#@2l 811# D 1/2

u1,1,0,1;~m!111111231231&

1ql 82s812p2 l 1s11S @ l 112 l 2311#@ l 312 l 4322#

@2l 11#@2l 811# D 1/2

u0,1,1,1;~m!111231231&J ,

~m!125a12

a11S @2l #@2l 11#@ l 132 l 1121#

@ l 112 l 2311# D 1/2

~m!112a12pS q

pD l 2s1 l 82s8 @2l 11#

@2l #

3H @ l 132 l 1122#S @2l #@ l 332 l 3112#

@ l 112 l 2311#@2l 811# D1/2

u1,1,1,0;~m!111111231&

1p2 l 1s12S @2l #@ l 132 l 1121#@ l 332 l 3112#

@2l 811# D 1/2

u1,0,1,1;~m!111231&

1ql 82s811@ l 132 l 1122#S @2l #@ l 312 l 4322#

@@ l 112 l 2311##@2l 811# D1/2

u1,1,0,1;~m!111111231231&

1ql 82s812p2 l 1s11S @2l #@ l 132 l 1121#@ l 312 l 4322#

@2l 811# D 1/2

u0,1,1,1;~m!111231231&J ,

~m!135a13

a11S @2l 8#@2l 811#@ l 332 l 3112#

@ l 312 l 4322# D 1/2

~m!112a13pS q

pD l 2s1 l 82s8 @2l 811#

@2l 8#

3H @ l 332 l 3111#S @2l 8#@ l 132 l 1121#

@2l 11#@ l 312 l 4322# D1/2

u1,1,1,0;~m!111111231&

1p2 l 1s11@ l 332 l 3111#S @ l 112 l 231 l #@2l 8#

@2l 11#@ l 312 l 4322# D1/2

u1,0,1,1;~m!111231&

1ql 82s8S @ l 132 l 1121#@ l 332 l 3112#@2l 8#

@2l 11# D 1/2

u1,1,0,1;~m!111111231231&

1ql 82s811p2 l 1sS @ l 112 l 2311#@ l 332 l 3112#@2l 8#

@2l 11# D 1/2

u0,1,1,1;~m!111231231&J ,

6500 J. Math. Phys., Vol. 41, No. 9, September 2000 Nguyen Anh Ky

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~m!145a14

a11S @2l #@2l 11#@ l 132 l 1121#@2l 8#@2l 811#@ l 332 l 3112#

@ l 112 l 2311#@ l 312 l 4322# D 1/2

~m!11

2a14

a12S @2l 8#@2l 811#@ l 332 l 3112#

@ l 312 l 4322# D 1/2

~m!122a14

a13S @2l #@2l 11#@ l 132 l 1121#

@ l 112 l 2311# D 1/2

~m!13

2a14pS q

pD l 2s1 l 82s821 @2l 11#@2l 811#

@2l #@2l 8# H @ l 132 l 1122#@ l 332 l 3111#

3S @2l #@2l 8#

@ l 112 l 2311#@ l 312 l 4322# D1/2

u1,1,1,0;~m!111111231&1p2 l 1s12@ l 332 l 3111#

3S @ l 132 l 1121#@2l #@2l 8#

@ l 312 l 4322# D 1/2

u1,0,1,1;~m!111231&1ql 82s8@ l 132 l 1122#

3S @2l #@2l 8#@ l 332 l 3112#

@ l 112 l 2311# D 1/2

u1,1,0,1;~m!111111231231&

1ql 82s811p2 l 1s11~@2l #@ l 132 l 1121#@ l 332 l 3112#@2l 8# !1/2u0,1,1,1;~m!111231231&J ,

~m!155a15~m!,

where l 5 12 (m132m23), s5m112

12 (m131m23), l 85 1

2 (m332m43), ands85m31212 (m331m43),

while (m)k6 i j is a QGZ basis vector obtained from (m)k with replacing the elementmi j by mi j

61. We can write down the coefficients in~3.27! all in terms ofl, s, l 8, ands8 only, but here weleave them partially expressed in terms ofl i j andl i j8 . From~3.27! we can immediately find all the~local! lowest weight vectors (M )k

V which, by definition, are annihilated byE21 andE43. Let usremind the reader again that every firdirmodVk

p,q on a levelh, spanned on linear combinations ofuu1 ,u2 ,u3 ,u4 ;(m)& in ~3.14! with a fixed ( i 51

4 u i[h is a linear space of homogeneous polyno-mials of a definite powerh in the negative odd generatorsEi j (1< j ,3< i<4) acting on (m)PV0

p,q(@m#). Taking into account all results obtained above we have proved the following asser-tion.

Proposition 3: EveryUp,q@gl(2/2)0#-fidirmod Vkp,q in decomposition (3.16) is characterized by

a highest weight@m#k given in (3.25) and is spanned by a QGZ basis(m)k given in (3.27).The latest formula~3.27!, in fact, represents a way in which the reduced basis is expressed in

terms of the induced basis and vice versa it is not a problem for us to find the inverse relationbetween these bases~see the Appendix!. For further convenience the vectors (m) k[(m)k ~for k56, 7, 9, 10, 12, 13, and 14! are partially given via other (m)k which are completely expressed interms ofuu1 ,u2 ,u3 ,u4 ;(m)&. It is not difficult to write down the explicit decompositions of these(m) k in the induced basis. But here we prefer the expressions in~3.27! which are more compactand more convenient for finding the inverse relation between two bases and matrix elements ofodd generators.

Now we are ready to calculate the matrix elements of the generatorsEi j . It is sufficient tocalculate the matrix elements of the Cartan–Chevalley generators only, since Up,q@gl(2/2)# can begenerated by these generators and any of its representation in some basis is completely defined bytheir actions on the same basis. For the even generators which do not shift theh’s we readily have

E11~m!k5~ l 1111!~m!k ,

E22~m!k5~ l 121 l 222 l 1112!~m!k ,

E12~m!k5~@ l 122 l 11#@ l 112 l 22# !1/2~m!k111,

6501J. Math. Phys., Vol. 41, No. 9, September 2000 Induced representations of the two-parametric . . .

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E21~m!k5~@ l 122 l 1111#@ l 112 l 2221# !1/2~m!k211,

L1~m!k5 12 ~ l 122 l 2221!~m!k ,

~3.28!E33~m!k5~ l 3111!~m!k ,

E44~m!k5~ l 321 l 422 l 3112!~m!k ,

E34~m!k5~@ l 322 l 31#@ l 312 l 42# !1/2~m!k131,

E43~m!k5~@ l 322 l 3111#@ l 312 l 4221# !1/2~m!k231,

L3~m!k5 12 ~ l 322 l 4221!~m!k .

As the matrix elements ofE23 andE32 are very long expressions we only explain here how to findthem. By construction a reduced basis vector (m)k in ~3.27! belonging to a fidirmodVk

p,q on alevel h is a homogeneous polynomial of a powerh in odd generatorsEi j , 1< j ,3< i<4, actingon (m)PV0

p,q . Under the action ofE23 ~or E32, respectively! this vector (m)k is shifted to otherfidirmodsVk8

p,q on the previous levelh21 ~or on the next levelh11, respectively!, i.e., we get onthe rhs ofE23(m)k @or E32(m)k , respectively# a homogeneous polynomial of a degreeh21 ~orh11, respectively!. Using the inverse relations~A1! we can express the latter polynomials ob-tained in terms of the reduced basis, that is, we get matrix elements ofE23 andE32 in this basis.It is a standard way to find matrix elements but in practice we can use a trick making calculationssimpler. SinceE23 commutes withE21 andE43 we first calculate the action ofE23 on the highestvectors only and then apply~3.11! to find all matrix elements of this generator on arbitrary (m)k .It is less complicated to compute matrix elements ofE32 in the standard way, but we can apply asimilar trick, namely, we first calculate the action ofE23 ~which commutes withE12 andE34! onthe lowest weight vectors and then apply the rule inverse to~3.11!.

It can be shown that the representations constructed contain all finite-dimensional irreduciblerepresentations of Up,q@gl(2/2)# classified as typical or nontypical representations which are sub-jects of next investigations.

IV. CONCLUSION

We have considered the two-parametric quantum deformations Up,q@gl(2/2)# and described indetail a method for constructing its finite-dimensional representations. The representations con-structed can be decomposed into finite-dimensional irreducible representations of the even subal-gebra Up,q@gl(2/2)0# and therefore can be given in bases of the latter. Using Poincare´–Birkhoff–Witt theorem and the induced representation method we constructed the induced basis of theinduced moduleWp,q. This basis, however, does not allow a clear description of a decompositionof Wp,q into Up,q@gl(2/2)0#-fidirmods. It was the reason the reduced basis was introduced. Thelatter basis representing a union of GZ bases of the even subalgebra Up,q@gl(2/2)0# according tothe branching rule Up,q@gl(2/2)#.Up,q@gl(2/2)0#.gl~1!^gl~1! is refered to as quasi-GZ basis.This step is intermediate but of independent interest. Having these two bases, the induced and thereduced ones, and the relations between them we can find all matrix elements of finite-dimensionalrepresentations of Up,q@gl(2/2)#. It turn out that the representations constructed contain all finite-dimensional irreducible representations of Up,q@gl(2/2)# and can be classified into typical andnontypical representations which are subjects of later papers.

Looking at the basis transformations and the matrix elements we observe, even at genericdeformation parameters, some ‘‘anomalies’’ which are canceled out atp5q. It means that thefinite-dimensional representations of the two-parametric quantum superalgebra Uq@gl(2/2)# arenot simply trivial deformations from those of the classical Lie superalgebra gl~2/2! in the sense

6502 J. Math. Phys., Vol. 41, No. 9, September 2000 Nguyen Anh Ky

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that they cannot be found from classical analogs by putting quantum deformation brackets inappropriate places unlike many cases of one-parametric deformations. For example, the expres-sions

q

p@2l #@ l 132 l 11#2@2l 11#@ l 132 l 1121# ~4.1!

and

q

p@2l 8#@ l 332 l 31#2@2l 811#@ l 332 l 3121# ~4.2!

appearing in the basis transformations and matrix elements can be written in the forms

S q

p21D @2l #@ l 132 l 11#1S q

pD l 132 l 1121

@ l 112 l 23# ~4.18!

and

S q

p21D @2l 8#@ l 332 l 31#1S q

pD l 332 l 3121

@ l 312 l 43#, ~4.28!

respectively. Atp5q the latest expressions become@ l 112 l 23# and@ l 312 l 43#, respectively, exactlyas in the one-parametric case.22,23

We hope that it is not very difficult to extend the present method to the case of one or bothdeformation parameters being roots of unity. For conclusion, let us emphasize that our method hasan advantage that it avoids the use of the Clebsch–Gordan coefficients which are not alwaysknown, especially for higher rank~classical and quantum! groups and multi-parametric deforma-tions.

ACKNOWLEDGMENTS

I would like to thank the Nishina memorial foundation for financial support and the Depart-ment of Physics, Chuo University, Tokyo, Japan for warm hospitality. Fruitful discussions with K.Furuta, T. Inami, and other members of the Theory Group of the Department of Physics, ChuoUniversity, are also hereby acknowledged. This work was supported in part by the NationalResearch Program for Natural Sciences of Vietnam under Grant No. KT-04.1.1.

APPENDIX

The induced basis~4.20! is expressed in terms of the reduced basis through the followinginverse relation:

u1,0,0,0;~m!&521

a1ql 1s21p2 l 82s8S @ l 132 l 1111#@ l 332 l 3111#

@2l 11#@2l 811# D 1/2

~m!1211

21

a2

q2 l 1s21p2 l 82s821

@2l 11# S @2l #@ l 112 l 2321#@ l 332 l 3111#

@2l 811# D 1/2

~m!2211

21

a3

ql 1spl 82s8

@2l 811# S @ l 132 l 1111#@ l 312 l 4321#@2l 8#

@2l 11# D 1/2

~m!3211

11

a4

q2 l 1spl 82s821

@2l 11#@2l 811#~@2l #@ l 112 l 2321#@2l 8#@ l 312 l 4321# !1/2~m!4

211,

6503J. Math. Phys., Vol. 41, No. 9, September 2000 Induced representations of the two-parametric . . .

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u0,1,0,0;~m!&521

a1ql 1sS @ l 132 l 1111#@ l 312 l 43#

@2l 11#@2l 811# D 1/2

~m!1211131

21

a2

q2 l 1s

p@2l 11# S @2l #@ l 112 l 2321#@ l 312 l 43#

@2l 811# D 1/2

~m!2211131

11

a3S p

qD l 82s821 ql 1s

@2l 811# S @ l 132 l 1111#@2l 8#@ l 332 l 31#

@2l 11# D 1/2

~m!3211131

21

a4S p

qD l 82s822 q2 l 1s21

@2l 11#@2l 811# S @2l #@ l 112 l 2321#@2l 8#

@ l 332 l 31#D 1/2

~m!4211131,

u0,0,1,0;~m!&521

a1q21p2 l 82s8S @ l 112 l 23#@ l 332 l 3111#

@2l 11#@2l 811# D 1/2

~m!1

11

a2S p

qD l 2s p2 l 82s821

@2l 11# S @2l #@ l 132 l 11#@ l 332 l 3111#

@2l 811# D 1/2

~m!2

11

a3

pl 82s8

@2l 811# S @ l 112 l 23#@2l 8#@ l 312 l 4321#

@2l 11# D 1/2

~m!3

11

a4S p

qD l 2s21 pl 82s8

@2l 11#@2l 811#~@2l #@ l 132 l 11#@2l 8#@ l 312 l 4321# !1/2~m!4 ,

u0,0,0,1;~m!&51

a1S @ l 112 l 23#@ l 312 l 4311#

@2l 11#@2l 811# D 1/2

~m!1131

21

a2S p

qD l 2s21 1

@2l 11# S @2l #@ l 132 l 11#@ l 312 l 43#

@2l 811# D 1/2

~m!2131

21

a3S p

qD l 82s821 1

@2l 811# S @ l 112 l 23#@2l 8#@ l 332 l 31#

@2l 11# D 1/2

~m!3131

21

a4S p

qD l 2s1 l 82s822 1

@2l 11#@2l 811# S @2l #@2l 8#

@ l 132 l 11#@ l 312 l 43#D 1/2

~m!4131,

u1,1,0,0;~m!&51

a5q2~ l 1s!S p

qD l 82s8S @ l 132 l 1111#@ l 132 l 1112#

@2l 11#@2l 12# D 1/2

~m!5211211131

11

a6S p

qD l 82s823 q22~ l 2s!23

@2l 12#~p2q2l 211~p2q!@2l 21# !~@ l 112 l 2321#

3@ l 132 l 1121# !1/2~m!62112111312

1

a7S p

qD l 82s823 q22~ l 2s11!

@2#@2l 11#@2l 12#

3~@2l #@2l 21#@ l 112 l 2322#@ l 112 l 2321# !1/2~m!7211211131,

E~22![u0,1,1,0;~m!&2pu1,0,0,1;~m!&

51

a5ql 1s11S p

qD l 82s811

@2#S @ l 132 l 1111#@ l 112 l 23#

@2l 11#@2l 12# D 1/2

~m!5211131

11

a6S p

qD l 2s1 l 82s822 q2 l 1s

@2l 12#$@2#@2l 21#@ l 132 l 11#2@2l #~p2@ l 132 l 11#

6504 J. Math. Phys., Vol. 41, No. 9, September 2000 Nguyen Anh Ky

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1q21@ l 132 l 1121# !%~m!62111311

1

a7S p

qD l 2s1 l 82s822 q2 l 1s

@2l 11#@2l 12#

3~@2l #@2l 21#@ l 132 l 11#@ l 112 l 2321!#1/2~m!7211131,

u0,0,1,1;~m!&51

a5S p

qD l 82s8S @ l 112 l 23#@ l 112 l 2311#

@2l 11#@2l 12# D 1/2

~m!51311

1

a6S p

qD 2~ l 2s!1 l 82s824

3H p

q@2l #@ l 132 l 1122#2@2l 21#@ l 132 l 1121#J 1

@2l 12# S @ l 132 l 11#

@ l 112 l 23#D 1/2

~m!6131

21

a7S p

qD 2~ l 2s!1 l 82s824 ~@2l #@2l 21#@ l 132 l 1121#@ l 132 l 11# !1/2

@2#@2l 11#@2l 12#~m!7

131,

u1,0,1,0;~m!&51

a8q21p22~ l 81s8!S p

qD 2 l 1sS @ l 332 l 3111#@ l 332 l 3112#

@2l 811#@2l 812# D 1/2

~m!8211

11

a9q21p2~ l 82s8!21S p

qD l 2s ~@ l 332 l 3111#@ l 312 l 4321# !1/2

@2l 812#~m!9

211

21

a10S p

qD l 2s qp2~ l 82s8!21

@2#@2l 811#@2l 812#

3~@2l 8#@2l 821#@ l 312 l 4322#@ l 312 l 4321# !1/2~m!10211,

~A1!

E~12![u0,1,1,0;~m!&1q21u1,0,0,1;~m!&

51

a8q21p2 l 82s821S p

qD l 2s11

@2#S @ l 332 l 3111#@ l 312 l 43#

@2l 811#@2l 812# D 1/2

~m!8211131

11

a9S p

qD l 2s1 l 82s822 pl 82s8

q@2l 812#$@2#@2l 21#@ l 332 l 31#2@2l 8#~q22@ l 332 l 31#

1p@ l 332 l 3121# !%~m!92111311

1

a10S p

qD l 2s1 l 82s822 pl 82s8

q@2l 811#@2l 812#

3~@2l #@2l 21#@ l 332 l 31#@ l 312 l 4321# !1/2~m!10211131,

u0,1,0,1;~m!&51

a8S p

qD l 2sS @ l 312 l 43#@ l 312 l 4311#

@2l 811#@2l 812# D 1/2

~m!82111311311

1

a9S p

qD l 2s12~ l 82s8!24

31

@2l 812# S @ l 332 l 31#

@2l 811# D1/2S p

q@2l 8#@ l 332 l 3122#2@2l 821#@ l 332 l 3121# D

3~m!92111311312

1

a10S p

qD l 2s12~ l 82s8!24 1

@2#@2l 811#@2l 812#

3~@2l 8#@2l 821#@ l 332 l 3121#@ l 332 l 31# !1/2~m!10211131131,

6505J. Math. Phys., Vol. 41, No. 9, September 2000 Induced representations of the two-parametric . . .

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u1,1,1,0;~m!&5q2 l 1s22pl 82s822S q

pD l 2s1 l 82s8

3H 1

a11S @ l 132 l 1111#@ l 332 l 3111#

@2l 11#@2l 811# D 1/2

~m!11211211131

11

a12

q

@2l 11# S @2l #@ l 112 l 2321#@ l 332 l 3111#

@2l 811# D ~m!12211211131

11

a13

q2

@2l 811# S @ l 132 l 1111#@2l 8#@ l 312 l 4321#

@2l 11# D 1/2

~m!13211211131

11

a14

q3

@2l 11#@2l 811#~@2l #@ l 112 l 2321#@2l 8#@ l 312 l 4321# !1/2~m!14

211211131J ,

u1,0,1,1;~m!&5pl 82s8S p

qD 2~ l 2s!1 l 82s823H 1

a11q23~q@2l #@ l 132 l 11#2p@2l 11#@ l 132 l 1121# !

3~p@2l 811#2q2@2l 8# !S @ l 332 l 3111#

@2l 11#@2l 811#@ l 112 l 23#D 1/2

~m!11211131

21

a12

q22

@2l 11#~p@2l 811#2q2@2l 8# !S @2l #@ l 132 l 11#@ l 332 l 3111#

@2l 811# D 1/2

3~m!122111312

1

a13

q21

@2l 811#~q21@2l #@ l 132 l 11#2p@2l 11#@ l 132 l 1121# !

3S @2l 8#@ l 312 l 4321#

@2l 11#@ l 112 l 23#D 1/2

~m!132111311

1

a14

1

@2l 11#@2l 811#

3~@2l #@ l 132 l 11#@2l 8#@ l 312 l 4321# !1/2~m!14211131J ,

u1,1,0,1;~m!&5q2 l 1s23S p

qD l 2s12~ l 82s8!24H 1

a11~q@2l #2p2@2l 11# !~p@2l 811#@ l 332 l 3121#

2q@2l 8#@ l 332 l 31# !S @ l 132 l 1111#

@2l 11#@2l 811#@ l 312 l 43#D 1/2

~m!11211211131131

21

a12

q

@2l 11#~p@2l 811#@ l 332 l 3121#2q@2l 8#@ l 332 l 31# !

3S @2l #@ l 112 l 2321#

@2l 811#@ l 312 l 43#D 1/2

~m!122112111311311

1

a13

q

@2l 811#~q@2l #2p2@2l 11# !

3S @ l 132 l 1111#@2l 8#@ l 332 l 31#

@2l 11# D 1/2

~m!13211211131131

11

a14

q2

@2l 11#@2l 811#~@2l #@ l 112 l 2321#@2l 8#@ l 332 l 31# !1/2~m!14

211211131131J ,

6506 J. Math. Phys., Vol. 41, No. 9, September 2000 Nguyen Anh Ky

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u0,1,1,1;~m!&5S p

qD 2~ l 2s1 l 82s822!H 1

a11q22~p@2l 11#@ l 132 l 1121#2q@2l #@ l 132 l 11# !

3~p@2l 811#@ l 332 l 3121#2q@2l 8#@ l 332 l 31# !

3S 1

@2l 11#@2l 811#@ l 112 l 23#@ l 312 l 43#D 1/2

~m!112111311311

1

a12

q21

@2l 11#

3~p@2l 811#@ l 332 l 3121#2q@2l 8#@ l 332 l 31# !

3S @2l #@ l 132 l 11#

@2l 811#@ l 312 l 43#D 1/2

~m!122111311311

1

a13

q21

@2l 811#~p@2l 11#@ l 132 l 1121#

2q@2l #@ l 132 l 11# !S @2l 8#@ l 332 l 31#

@2l 11#@ l 112 l 23#D 1/2

~m!13211131131

21

a14

1

@211#@2l 811#~@2l #@ l 132 l 11#@2l 8#@ l 332 l 31# !1/2~m!14

211131131J ,

u1,1,1,1;~m!&51

a15~m!211211131131.

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6507J. Math. Phys., Vol. 41, No. 9, September 2000 Induced representations of the two-parametric . . .

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31A. H. Kamupingene, N. A. Ky, and D. Palev, J. Math. Phys.30, 553 ~1989!; T. Palev and N. Stoilova,ibid. 31, 953~1990!.

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6508 J. Math. Phys., Vol. 41, No. 9, September 2000 Nguyen Anh Ky

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