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FINITE-DIMENSIONAL REPRESENTATIONS

arXiv:hep-th/9305183v2 14 Nov 1994FINITE-DIMENSIONAL REPRESENTATIONSOF THE QUANTUM SUPERALGEBRA Uq[gl(2/2)]:I. Typical representations at generic qNguyen Anh Ky ∗)International Centre for Theoretical PhysicsP.O.

Box 586, Trieste 34100, ItalyAbstractIn the present paper we construct all typical finite-dimensional representationsof the quantum Lie superalgebra Uq[gl(2/2)] at generic deformation parameter q.As in the non-deformed case the finite-dimensional Uq[gl(2/2)]-module W q obtainedis irreducible and can be decomposed into finite-dimensional irreducible Uq[gl(2) ⊕gl(2)]-submodules V qk .PACS numbers:02.20Tw, 11.30Pb.∗)Permanent Mailing Address: Centre for Polytechnic Educations (Trung tˆam gi´ao du.c k˜y thuˆa.ttˆong ho.p), Vinh, VietnamAddress after June 01, 1993: Institute for Nuclear Research and Nuclear Energy, Boul. Tsari-gradsko chaussee 72, Sofia 1784, Bulgaria

1. IntroductionSince the quantum deformations 1−5 became a subject of intensive investigationsmany (algebraic and geometric) structures and different representations of quantum(super-) groups have been obtained and understood.

For instance, the quantum al-gebra Uq[sl(2)] is very well studied 6−8. Originated from intensive investigations onthe quantum inverse scattering method and the Yang-Baxter equations, the quan-tum groups have found various applications in theoretical physics and mathematics(see in this context, for example, Refs.9−12).As in the non-deformed case forapplications of quantum groups we often need their explicit representations.

Be-ing a subject of many investigations, representations of quantum groups, especiallyrepresentations of quantum superalgebras are presently under development. How-ever, although the progress in this direction is remarkable the problem is still farfrom being satisfactorily solved.

Explicit representations are known only for quan-tum Lie superalgebras of lower ranks and of particular types like Uq[osp(1/2)] 13,Uq[gl(1/n)] 14, etc.. For higher rank quantum Lie superalgebras 15−18, besides someq-oscillator representations which are most popular among those constructed, we donot know so much about other representations, in particular the finite-dimensionalones. Some general aspects and module structures of finite-dimensional representa-tions of Uq[gl(m/n)] are considered in Ref.18 (see also Ref.14) but without theirexplicit constructions.

So the question concerning an explicit construction of finite-dimensional representations of Uq[gl(m/n)] is still unsolved for m and n ≥2.Here, extending the method developed by Kac 19 in the case of Lie superalgebras(from now on, only superalgebras) we shall construct all finite-dimensional represen-tations of the quantum Lie superalgebra Uq[gl(2/2)] at generic q, i.e. q is not a rootof unity.

It turns out that the finite-dimensional Uq[gl(2/2)]-modules have similarstructures to that of the non-deformed ones 20,21 and are decomposed into finite-dimensional irreducible Uq[gl(2) ⊕gl(2)]-modules. Finite-dimensional Uq[gl(2/2)]-modules can be classified again as typical or atypical ones (see the Proposition 2).In the frame-work of this paper for the sake of simplicity we shall consider only thetypical representations at generic q.

When q is a root of unity, as emphasized also

in 18, the structures of Uq[gl(2/2)]-modules are drastically different in comparisonwith the structures of gl(2/2)-modules 20,21. The present investigation on typicalrepresentations at generic q is easily extended on atypical representations at genericq 22 and finite-dimensional representations at q being a root of unity 23.The paper is organized as follows.

In order to make the present constructionclear, in the section 2 we expose some introductory concepts and basic definitionsof quantum superalgebras, especially Uq[gl(m/n)]. We also describe briefly the pro-cedure used for constructing finite-dimensional representations of Uq[gl(m/n)].

Thequantum superalgebra Uq[gl(2/2)] is defined in section 3. The section 4 is devotedto construction of finite-dimensional representations of Uq[gl(2/2)].

Some commentsand the conclusion are made in the section 5, while the references are given in thelast section 6.For a convenient reading we shall keep as many as possible the abbreviationsand notations used in Ref. 20 among the following ones:fidirmod(s) - finite-dimensional irreducible module(s),GZ basis - Gel’fand-Zetlin basis,lin.env.

{X} - linear envelope of X,q - the deformation parameter,V ql ⊗V qr - tensor product between two linear spaces V ql and V qror a tensor product between a Uq[gl(2)l]-module V qland a Uq[gl(2)r]-module V qr ,T q ⊙V q0 - tensor product between two Uq[gl(2) ⊕gl(2)]-modulesT q and V q0 ,[x]q = qx−q−xq−q−1 , where x is some number or operator,[x] ≡[x]q2,[E, F} - supercommutator between E and F,[E, F]q ≡EF −qFE - q-deformed commutator between E and F,

aij - an element of the Cartan matrix (aij),qi = qdi, where di are rational numbers such thatdiaij = djaji, i, j = 1, 2, ..., r,Ei = eiq−hii≡eik−1i ki+1,Fi = fiq−hii≡fik−1i ki+1.Note that we must not confuse the quantum deformation [x] ≡[x]q2 of x with thehighest weight (signature) [m] in the GZ basis (m) or with the notation [ , ] forcommutators.2. Some introductory concepts of quantum superalgebrasLet g be a rank r (semi-) simple superalgebra, for example, sl(m/n) or osp(m/n).The quantum superalgebra Uq(g) as a quantum deformation (q-deformation) of theuniversal enveloping algebra U(g) of g, is completely defined by the Cartan-Chevalleycanonical generators hi, ei and fi, i = 1, 2, ..., r which satisfy 15−17a) the quantum Cartan-Kac supercommutation relations[hi, hj]= 0,[hi, ej]= aijej,[hi, fj]= −aijfj,[ei, fj}= δij[hi]q2i ,(2.1)b) the quantum Serre relations(adqEi)1−˜aijEj = 0,(adqFi)1−˜aijFj = 0(2.2)where (˜aij) is a matrix obtained from the non-symmetric Cartan matrix (aij) byreplacing the strictly positive elements in the rows with 0 on the diagonal entry by−1, while adq is the q-deformed adjoint operator given by the formula (2.8)and

c) the quantum extra-Serre relations 24−26 (for g being sl(m/n) or osp(m/n)){[em−1, em]q2, [em, em+1]q2} = 0,{[fm−1, fm]q2, [fm, fm+1]q2} = 0,(2.3)being additional constraints on the unique odd Chevalley generators em and fm. Inthe above formulas we denoted qi = qdi where di are rational numbers symmetrizingthe Cartan matrix diaij = djaji, 1 ≤i, j ≤r.

For example, in case g = sl(m/n) wehavedi =1if 1 ≤i ≤m,−1if m + 1 ≤i ≤r = m + n −1. (2.4)The above-defined quantum superalgebras form a subclass of a special class ofHopf algebras called by Drinfel’d quasitriangular Hopf algebras 2.

They are endowedwith a Hopf algebra structure given by the following additional maps:a) coproduct ∆:U →U ⊗U∆(1) = 1 ⊗1,∆(hi)= hi ⊗1 + 1 ⊗hi,∆(ei)= ei ⊗qhii + q−hii⊗ei,∆(fi)= fi ⊗qhii + q−hii⊗fi,(2.5)b) antipode S :U →US(1) = 1,S(hi)= −hi,S(ei)= −qaiii ei,S(fi)= −q−aiiifi(2.6)andc) counit ε : U →C

ε(1) = 1,ε(hi) = ε(ei) = ε(fi) = 0,(2.7)Then the quantum adjoint operator adq has the following form 16,27adq = (µL ⊗µR)(id ⊗S)∆(2.8)with µL (respectively, µR) being the left (respectively, right) multiplication: µL(x)y =xy (respectively, µR(x)y = (−1)degx.degyyx).A quantum superalgebra Uq[gl(m/n)] is generated by the generators k±1i≡q±Eiii,ej ≡Ej,j+1 and fj ≡Ej+1,j, i = 1, 2, ..., m + n, j = 1, 2, ..., m + n −1 such that thefollowing relations hold (cf. Refs.

14,18)a) the super-commutation relationskikj= kjki ,kik−1i=k−1i ki = 1 ,kiejk−1i= q(δij−δi,j+1)iej ,kifjk−1i=q(δij+1−δi,j)ifj ,[ei, fj} = δij[hi]q2i , whereqhii=kik−1i+1,(2.9)b) the Serre relations (2.2) taking now the explicit forms[ei, ej] = [fi, fj] = 0, if |i −j| ̸= 1,e2m=f 2m= 0,[ei, [ei, ej]q±2]q∓2 = [fi, [fi, fj]q±2]q∓2 = 0,if |i −j| = 1(2.10)andc) the extra-Serre relations (2.3){[em−1, em]q2, [em, em+1]q2} = 0,{[fm−1, fm]q2, [fm, fm+1]q2} = 0. (2.11)Here, besides di, 1 ≤i ≤r = m + n −1 given in (2.4) we introduced dm+n = −1.The Hopf structure on ki looks as

∆(ki) = ki ⊗ki,S(ki) = k−1i ,ε(ki) = 1. (2.12)The generators Eii, Ei,i+1 and Ei+1,i together with the generators defined in thefollowing wayEij:= [EikEkj]q−2 ≡EikEkj −q−2EkjEik, i < k < j,Eji:= [EjkEki]q2 ≡EjkEki −q2EkiEjk,i < k < j,(2.13)play an analogous role as the Weyl generators eij,(eij)kl = δikδjl,(2.14)of the superalgebra gl(m/n) whose universal enveloping algebra U[gl(m/n)] repre-sents a classical limit of Uq[gl(m/n)] when q →1.The quantum algebra Uq[gl(m/n)0] ∼= Uq[gl(m) ⊕gl(n)] generated by ki, ej andfj, i = 1, 2, ..., m + n, m ̸= j = 1, 2, ..., m + n −1,Uq[gl(m/n)0] = lin.env.

{Eij∥1 ≤i, j ≤m and m + 1 ≤i, j ≤m + n}(2.15)is an even subalgebra of Uq[gl(m/n)]. Note that Uq[gl(m/n)0] is included in thelargest even subalgebra Uq[gl(m/n)]0 containing all elements of Uq[gl(m/n)] witheven powers of the odd generators.As is shown by M. Rosso 28 and C. Lusztig 29, a finite-dimensional represen-tation of a Lie algebra g can be deformed to a finite-dimensional representationof its quantum analogue Uq(g).In particular, finite-dimensional representationsof Uq[gl(m) ⊕gl(n)] are quantum deformations of those of gl(m) ⊕gl(n).

Hence,a finite-dimensional irreducible representation of Uq[gl(m) ⊕gl(n)] is again high-est weight.Following the classical procedure 19,20 we can construct representa-tions of Uq[gl(m/n)] induced from finite-dimensional irreducible representations ofUq[gl(m) ⊕gl(n)] which, as we can see from (2.9-11) and (2.15), is the stabilitysubalgebra of Uq[gl(m/n)]. Let V q0 (Λ) be a Uq[gl(m) ⊕gl(n)]-fidirmod characterized

by some highest weight Λ. For a basis of V q0 (Λ) we can choose the Gel’fand-Zetlin(GZ) tableaux 30, since the latter are invariant under the quantum deformations28,29,31,32.

Therefore, the highest weight Λ is described again by the first row of theGZ tableaux called from now on as the GZ (basis) vectors.DemandingEm,m+1V q0 (Λ) ≡emV q0 (Λ) = 0(2.16)i.e.Uq(A+)V q0 (Λ) = 0(2.17)we turn V q0 (Λ) into a Uq(B)-module, whereA+ = {Eij∥1 ≤i ≤m < j ≤m + n}(2.18)B = A+ ⊕gl(m) ⊕gl(n)(2.19)The Uq[gl(m/n)] -module W q induced from the Uq[gl(m) ⊕gl(n)]-module V q0 (Λ) isthe factor-spaceW q = W q(Λ) = [Uq ⊗V q0 (Λ)]/Iq(Λ)(2.20)where Uq ≡Uq[gl(m/n)], while Iq(Λ) is the subspaceIq(Λ) = lin.env. {ub ⊗v −u ⊗bv∥u ∈Uq, b ∈Uq(B) ⊂Uq, v ∈V q0 (Λ)}(2.21)In order to complete the present section let us note that the modules W q(Λ) andV q0 (Λ) have one and the same highest vector.

Therefore, they are characterized byone and the same highest weight Λ.3.U[gl(2/2)]U[gl(2/2)]The quantum superalgebra Uq[gl(2/2)]The quantum superalgebra Uq ≡Uq[gl(2/2)] is generated by the generators Eii,i = 1, 2, 3, 4, E12 ≡e1, E23 ≡e2, E34 ≡e3, E21 ≡f1, E32 ≡f2 and E43 ≡f3satisfying the relations (2.9-11) which now reada) the super-commutation relations (1 ≤i, i + 1, j, j + 1 ≤4):[Eii, Ejj]= 0,

[Eii, Ej,j+1]= (δij −δi,j+1)Ej,j+1,[Eii, Ej+1,j]= (δi,j+1 −δij)Ej+1,j,[Ei,i+1, Ej+1,j}= δij[hi]q2,hi = (Eii −di+1di Ei+1,i+1),(3.1)with d1 = d2 = −d3 = −d4 = 1,b) the Serre-relations:[E12, E34]= [E21, E43]= 0,E223=E232= 0,[E12, E13]q2= [E24, E34]q2 = 0,[E21, E31]q2= [E42, E43]q2 = 0,(3.2)andc) the extra-Serre relations:{E13, E24} = 0,{E31, E42} = 0,(3.3)respectively. Here, for a further convenience, the operatorsE13 := [E12, E23]q−2,E24 := [E23, E34]q−2,E31 := −[E21, E32]q−2,E42 := −[E32, E43]q−2.

(3.4)and the operators composed in the following wayE14 :=[E12, [E23, E34]q−2]q−2 ≡[E12, E24]q−2,E41 :=[E21, [E32, E43]q−2]q−2 ≡−[E21, E42]q−2(3.5)are defined as new generators. The latter are odd and have vanishing squares.

They,together with the Cartan-Chevalley generators, form a full system of q-analogues ofthe Weyl generators eij, 1 ≤i, j ≤4, of the superalgebra gl(2/2) whose universalenveloping algebra U[gl(2/2)] is a classical limit of Uq[gl(2/2)] when q →1. Othercommutation relations between Eij follow from the relations (3.1-3) and the defini-

tions (3.4-5).The subalgebra Uq[gl(2/2)0] (⊂Uq[gl(2/2)]0 ⊂Uq[gl(2/2)]) is even and isomor-phic to Uq[gl(2) ⊕gl(2)] ≡Uq[gl(2)] ⊕Uq[gl(2)] which is completely defined by Eii,1 ≤i ≤4, E12, E34, E21 and E43Uq[gl(2/2)0] = lin.env. {Eij∥i, j = 1, 2 and i, j = 3, 4}(3.6)In order to distinguish two components of Uq[gl(2/2)0] we setleft Uq[gl(2)] ≡Uq[gl(2)l] := lin.env.

{Eij∥i, j = 1, 2},(3.7)right Uq[gl(2)] ≡Uq[gl(2)r] := lin.env. {Eij∥i, j = 3, 4}.

(3.8)That meansUq[gl(2/2)0] = Uq[gl(2)l ⊕gl(2)r]. (3.9)Let V q(Λ) be a Uq[gl(2/2)0]-fidirmod of the highest weight Λ.Thus V q can bedecomposed into a tensor productV q(Λ) = V ql (Λl) ⊗V qr (Λr),(3.10)between a Uq[gl(2)l]-fidirmod V ql (Λl) of a highest weight Λl and a Uq[gl(2)r]-fidirmodV qr (Λr) of a highest weight Λr, where Λl and Λr are defined respectively as the leftand right components of Λ :Λ = [Λl, Λr].(3.11)4.

Finite-dimensional representations of Uq[gl(2/2)]Here, we shall construct finite-dimensional representations of Uq[gl(2/2)] inducedfrom finite-dimensional irreducible representations of Uq[gl(2/2)0].In the frame-work of the present paper we consider only typical representations of Uq[gl(2/2)] atgeneric q. Atypical representations at generic q and finite-dimensional representa-tions of Uq[gl(2/2)] at roots of unity are subjects of later publications 22,23.

As mentioned earlier a fidirmod V q0 (Λ) of the quantum algebra Uq[gl(2/2)0] repre-sents a quantum deformation (q-deformation) of some fidirmod V0(Λ) of the algebragl(2/2)0. Moreover, following the classical procedure we can construct Uq[gl(2/2)]-fidirmods induced from Uq[gl(2/2)0]-fidirmods.

SettingE23V q0 = 0,(4.1)a Uq[gl(2/2)]-module W q induced from the Uq[gl(2/2)0]-fidirmod V q0 , by the con-struction, is the factor-space (2.20) with m = n = 2 :W q(Λ) = [Uq ⊗V q0 (Λ)]/Iq(Λ),(4.2)whereIq(Λ) = lin.env. {ub ⊗v −u ⊗bv∥u ∈Uq, b ∈Uq(B) ⊂Uq, v ∈V q0 (Λ)}Uq(B) = lin.env.

{Eij, E23∥i, j = 1, 2 and i, j = 3, 4}(4.3)Any vector w from the module W q has the formw = u ⊗v,u ∈Uq,v ∈V q0(4.4)Then W q is a Uq[gl(2/2)]-module in the sensegw ≡g(u ⊗v) = gu ⊗v ∈W q(4.5)for g, u ∈Uq, w ∈W q and v ∈V q0 .In the next two subsections we shall construct the bases of the module W q andfind the explicit matrix elements for the typical representations of Uq[gl(2/2)].4.1 The basesSince the GZ basis is invariant under the q-deformation, for a basis of a Uq[gl(2)]-fidirmod V0 we can choosem12m22m11≡[m]m11(4.6)

where mij are complex numbers such that m12 −m11 ∈Z+ and m11 −m22 ∈Z+.Under the actions of the Uq[gl(2)]-generators Eij, i, j = 1, 2 the basis (4.6) transformsas follows 32E11m12m22m11=(l11 + 1)m12m22m11,E22m12m22m11=(l12 + l22 −l11 + 2)m12m22m11,E12m12m22m11=([l12 −l11][l11 −l22])1/2m12m22m11 + 1,E21m12m22m11=([l12 −l11 + 1][l11 −l22 −1])1/2m12m22m11 −1,(4.7)where lij = mij −i for i = 1, 2 and lij = mij −i + 2 for i = 3, 4.On the other hands, V q0 is decomposed into the tensor productV q0 = V q0,l ⊗V q0,r. (4.8)where V q0,l and V q0,r are a Uq[gl(2)l]- and a Uq[gl(2)r]-fidirmods, respectively.

There-fore, the GZ basis of V q0 is the tensor productm13m23m11⊗m33m43m31≡[m]lm11⊗[m]rm31≡(m)l ⊗(m)r ≡(m) (4.9)between the GZ basis of V q0,l spanned on the vectors (m)l and the GZ basis of V q0,rspanned on the vectors (m)r. Following the approach of Ref.20 and keeping thenotations used there, we can represent the GZ basis (4.9) of V q0 in the formm13m23m11; m33m43m31≡[m]lm11; [m]rm31≡(m)(4.10)Then, the highest weight Λ is given by the first row (signature) [m13, m23, m33, m43] ≡[[m]l, [m]r] ≡[m] common for all the GZ basis vectors (4.10) of V q0 :V q0 ≡V q0 (Λ) = V q0 ([m]) = V q0,l([m]l) ⊗V q0,r([m]r)(4.11)

The explicit action of Uq[gl(2/2)0] on V q0 ([m]) follows directly from (4.7) and :g0(m) = g0,l(m)l ⊗(m)r + (m)l ⊗g0,r(m)r(4.12)for g0 ≡g0,l ⊕g0,r ∈Uq[gl(2/2)0] and (m) ∈V q0 ([m]).The GZ basis vectorm13m23m13; m33m43m33≡[m]lm13; [m]rm33≡(M)(4.13)satisfying the conditionsEii(M) = mi3(M),i = 1, 2, 3, 4,E12(M)= E34(M) = 0(4.14)by definition, is the highest weight vector in V q0 ([m]). Therefore, as in the classicalcase (q = 1) the highest weight [m] is nothing but an ordered set of the eigen valuesof the Cartan generators Eii on the highest weight vector (M).

The latter is alsohighest weight vector in W q([m]) because of the condition (4.1).All other, i.e.lower weight, basis vectors of V q0 can be obtained from the highest weight vector(M) through acting on the latter by monomials of definite powers of the loweringgenerators E21 and E43:(m)= [m11 −m23]! [m31 −m43]!

[m13 −m23]! [m13 −m11]!

[m33 −m43]! [m33 −m31]!

!1/2×(E21)m13−m11(E43)m33−m31(M),(4.15)where [n]’s are short hands ofq2n −q−2nq2 −q−2≡[n]q2 ≡[n],(4.16)while[n]! = [1][2]...[n −1][n].

(4.17)Using (3.1-5) we can show that a q-analogue of the Poincar´e-Birkhoff-Witt the-orem holds 26 (see also 18), namely Uq is a linear span of the elements of the formg = (E31)θ1(E32)θ2(E41)θ3(E42)θ4b,b ∈Uq(B),θi = 0, 1,i = 1, 2, 3, 4. (4.18)

Indeed, a right-ordered basis vector of Uq likeh = (E23)η1(E24)η2(E13)η3(E14)η4(E41)θ1(E31)θ2(E42)θ3(E32)θ4h0,where ηi, θi = 0, 1, h0 ∈Uq[gl(2/2)0], can be re-ordered and expressed through thevectors (4.18) which are more convenient for consulting the classical case in Refs.20,21. Taking into account the fact that V q0 ([m]) is a Uq(B)-module we haveW q([m]) = lin.env.

{(E31)θ1(E32)θ2(E41)θ3(E42)θ4 ⊗v∥v ∈V q0 , θ1, ..., θ4 = 0, 1}. (4.19)Consequently, the vectors|θ1θ2, θ3, θ4; (m)⟩:= (E31)θ1(E32)θ2(E41)θ3(E42)θ4 ⊗(m),(4.20)all together span a basis of the module W q([m]).

We shall call this basis induced inorder to distinguish it from the introduced later reduced basis and is more conve-nient for us to investigate the reducibleness of W q([m]).The subspace T q consisting of|θ1, θ2, θ3, θ4⟩:= (E31)θ1(E32)θ2(E41)θ3(E42)θ4(4.21)can be considered as a Uq[gl(2/2)0]-adjoint module (upto rescaling by ki in definitepowers). The latter is 16-dimensional as begins from |0, 0, 0, 0⟩when ∀θi = 0 andends at |1, 1, 1, 1⟩when ∀θi = 1.

Therefore, W q([m]), as a Uq[gl(2/2)0]-moduleW q([m]) = T q ⊙V q0 ([m]),(4.19′)is reducible and can be decomposed into 16 finite-dimensional irreducible Uq[gl(2/2)0]-submodules V qk ([m]k), k = 0, 1, ..., 15 :W q([m]) =15Mk=0V qk ([m]k). (4.22)Here, [m]k ≡[m12, m22, m32, m42]k are the local highest weights of the submodulesV qk in their GZ bases denoted now asm12m22m11; m32m42m31k≡(m)k.(4.23)

The highest weight [m]0 ≡[m] of V q0 being also the highest weight of W q is referredas a global highest weight. We call [m]k, k ≥1 the local highest weights in the sensethat they characterize only the submodules V qk ⊂W q as Uq[gl(2/2)0]-fidirmods,while the global highest weight [m] characterizes the Uq[gl(2/2)]-module W q as thewhole.

In the same way we define the local highest weight vectors (M)k in V qk asthose (m)k satisfying the conditions (cf. (4.14))Eii(M)k=mi2(M)k,i = 1, 2, 3, 4,E12(M)k=E34(M)k = 0.

(4.24)The highest weight vector (M) of V q0 is also the global highest weight vector in W qfor which the condition (see (4.1))E23(M) = 0(4.25)and the conditions (4.24) simultaneously hold.Let us denote by Γqk the basis system spanned on the basis vectors (m)k (4.23)in each V qk ([m]). For a basis of W q we can choose the union Γq = S15k=0 Γqk of all thebases Γqk, namely, a basis vector of W q has to be identified with one of the vectors(m)k, k = 0, 1, ..., 15.

The basis Γq is referred as a (Uq[gl(2/2)0]-) reduced basis. It isclear that every basis Γqk = Γk([m]k)q is labelled by a local highest weight [m]k, whilethe basis Γq = Γq([m]) is labelled by the global highest weight [m].

Going ahead, wemodify the notation (4.23) for the basis vectors in Γq as follows (see (3.54) in Ref.20)m13m23m33m43m12m22m32m42m110m310k≡m12m22m11; m32m42m31k≡(m)k,(4.26)with k running from 0 to 15 as for k = 0 we have to take into account mi2 = mi3,i = 1, 2, 3, 4, i.e. (m)0 ≡(m) =m13m23m33m43m13m23m33m43m110m310.

(4.27)

In (4.26) the first row [m] = [m13, m23, m33, m43] being the (global) highest weight ofW q is fixed for all the vectors in the whole W q and characterizes the module itself,while the second row is a (local) highest weight of some submodule V qk and tellsus that the considered basis vector (m)k of W q belongs to this submodule in thedecomposition (4.22) corresponding to the branching rule Uq[gl(2/2)] ⊃Uq[gl(2/2)0].It is easy to see that the highest vectors (M)k in the notation (4.26) are(M)k =m13m23m33m43m12m22m32m42m120m320k,k = 0, 1, ...15. (4.28)The (global) highest weight vector (M) (4.13) is given now by(M) =m13m23m33m43m13m23m33m43m130m330.

(4.29)Let us denote by (m)±ijka GZ vector obtained from (m)k by replacing the elementmij of the latter by mij ± 1. We can prove that the highest weight vectors (M)kexpressed in terms of the induced basis (4.20) have the following explicit forms(M)0=a0 |0, 0, 0, 0; (M)⟩,a0 ≡1,(M)1=a1 |0, 1, 0, 0; (M)⟩,(M)2=a2n|1, 0, 0, 0; (M)⟩+ q4l[2l]−1/2 0, 1, 0, 0; (M)−11Eo,(M)3=a3n|0, 0, 0, 1; (M)⟩−q−4l′−2[2l′]−1/2 0, 1, 0, 0; (M)−31Eo,(M)4=a4n|0, 0, 1, 0; (M)⟩+ q4l[2l]−1/2 0, 0, 0, 1; (M)−11E−q−4l′−2[2l′]−1/2 1, 0, 0, 0; (M)−31E−q4l−4l′−2 ([2l][2l′])−1/2 0, 1, 0, 0; (M)−11−31Eo,(M)5=a5 |0, 1, 0, 1; (M)⟩,(M)6=a6n|1, 0, 0, 1; (M)⟩+ q−2 |0, 1, 1, 0; (M)⟩

+q4l(q2 + q−2)[2l]−1/2 0, 1, 0, 1; (M)−11Eo,(M)7=a7n|1, 0, 1, 0; (M)⟩+ q4l[2l]−1/2 1, 0, 0, 1; (M)−11E+q4l−2[2l]−1/2 0, 1, 1, 0; (M)−11E+q8l−4 q2 + q−21/2 ([2l][2l −1])−1/2 0, 1, 0, 1; (M)−11−11E,(M)8=a8 |1, 1, 0, 0; (M)⟩,(M)9=a9n|1, 0, 0, 1; (M)⟩−q2 |0, 1, 1, 0; (M)⟩−q−4l′(q2 + q−2)[2l′]−1/2 1, 1, 0, 0; (M)−31Eo,(M)10=a10n|0, 0, 1, 1; (M)⟩−q−4l′−4[2l′]−1/2 1, 0, 0, 1; (M)−31E+q−4l′−2[2l′]−1/2 0, 1, 1, 0; (M)−31E+q−8l′ q2 + q−21/2 ([2l′][2l′ −1])−1/2 1, 1, 0, 0; (M)−31−31E,(M)11=a11 |1, 1, 0, 1; (M)⟩,(M)12=a12n|1, 1, 1, 0; (M)⟩+ q4l[2l]−1/2 1, 1, 0, 1; (M)−11Eo,(M)13=a13n|0, 1, 1, 1; (M)⟩+ q−4l′−2[2l′]−1/2 1, 1, 0, 1; (M)−31Eo,(M)14=a14n|1, 0, 1, 1; (M)⟩+ q4l[2l]−1/2 0, 1, 1, 1; (M)−11E+q−4l′−2[2l′]−1/2 1, 1, 1, 0; (M)−31E+q4l−4l′−2 ([2l][2l′])−1/2 1, 1, 0, 1; (M)−11−31Eo,(M)15=a15 |1, 1, 1, 1; (M)⟩,(4.30)where l = 12(m13 −m23) and l′ = 12(m33 −m43), while ak = ak(q) are coefficientsdepending on q. Indeed, (M)k given in (4.30) form a set of all linear independentvectors satisfying the conditions (4.24).

For a further convenience, let us rescale thecoefficients ak as followsa0=c0 ≡1 ,a8=q−2c8 ,a1=−q−2c1 ,a9=−q−2 [2l′][2][2l′ + 2]!1/2c9 ,

a2=−q−2 [2l][2l + 1]!1/2c2 ,a10=q2 [2l′ −1][2l′ + 1]!1/2c10 ,a3= [2l′][2l′ + 1]!1/2c3 ,a11=q−2c11 ,a4= [2l][2l′][2l + 1][2l′ + 1]!1/2c4 ,a12=q−2 [2l][2l + 1]!1/2c12 ,a5=q−2c5 ,a13= [2l′][2l′ + 1]!1/2c13 ,a6=q−2 [2l][2][2l + 2]!1/2c6 ,a14= [2l][2l′][2l + 1][2l′ + 1]!1/2c14 ,a7=q−2 [2l −1][2l + 1]!1/2c7,a15=c15 ,(4.31)where ck = ck(q) are some other constants which may depend on q. Looking at(4.30) we easily identify the highest weights [m]k[m]0 =[m13, m23, m33, m43],[m]1 =[m13, m23 −1, m33 + 1, m43],[m]2 =[m13 −1, m23, m33 + 1, m43],[m]3 =[m13, m23 −1, m33, m43 + 1],[m]4 =[m13 −1, m23, m33, m43 + 1],[m]5 =[m13, m23 −2, m33 + 1, m43 + 1],[m]6 =[m13 −1, m23 −1, m33 + 1, m43 + 1]6,[m]7 =[m13 −2, m23, m33 + 1, m43 + 1],[m]8 =[m13 −1, m23 −1, m33 + 2, m43],[m]9 =[m13 −1, m23 −1, m33 + 1, m43 + 1]9,[m]10 =[m13 −1, m23 −1, m33, m43 + 2],[m]11 =[m13 −1, m23 −2, m33 + 2, m43 + 1],[m]12 =[m13 −2, m23 −1, m33 + 2, m43 + 1],[m]13 =[m13 −1, m23 −2, m33 + 1, m43 + 2],[m]14 =[m13 −2, m23 −1, m33 + 1, m43 + 2],

[m]15 =[m13 −2, m23 −2, m33 + 2, m43 + 2]. (4.32)In the latest formula (4.32), with the exception of [m]6 and [m]9 where a degenera-tion is present, we skip the subscript k in the r.h.s..

The proofs of (4.30) and (4.32)follow from direct computations.Using the rule (4.15) which now reads(m)k= [m11 −m22]! [m31 −m42]!

[m12 −m22]! [m12 −m11]!

[m32 −m42]! [m32 −m31]!

!1/2×(E21)m12−m11(E43)m32−m31(M)k(4.15′)we can find all the basis vectors (m)k :(m)0=|0, 0, 0, 0; (m)⟩,(m)1=c1q2(l′−p′) [l13 −l11][l31 −l43 −1][2l + 1][2l′ + 1]!1/2 1, 0, 0, 0; (m)+11−31E−q2(−l+p+l′−p′) [l11 −l23][l31 −l43 −1][2l + 1][2l′ + 1]!1/2 0, 1, 0, 0; (m)−31E+ [l13 −l11][l33 −l31 + 1][2l + 1][2l′ + 1]!1/2 0, 0, 1, 0; (m)+11E−q2(−l+p) [l11 −l23][l33 −l31 + 1][2l + 1][2l′ + 1]!1/2|0, 0, 0, 1; (m)⟩,(m)2=c2−q2(l′−p′) [l11 −l23][l31 −l43 −1][2l + 1][2l′ + 1]!1/2 1, 0, 0, 0; (m)+11−31E−q2(l+p+l′−p′+1) [l13 −l11][l31 −l43 −1][2l + 1][2l′ + 1]!1/2 0, 1, 0, 0; (m)−31E− [l11 −l23][l33 −l31 + 1][2l + 1][2l′ + 1]!1/2 0, 0, 1, 0; (m)+11E−q2(l+p+1) [l13 −l11][l33 −l31 + 1][2l + 1][2l′ + 1]!1/2|0, 0, 0, 1; (m)⟩,(m)3=c3q−2(l′+p′+1) [l13 −l11][l33 −l31 + 1][2l + 1][2l′ + 1]!1/2 1, 0, 0, 0; (m)+11−31E

−q−2(l−p+l′+p′+1) [l11 −l23][l33 −l31 + 1][2l + 1][2l′ + 1]!1/2 0, 1, 0, 0; (m)−31E− [l13 −l11][l31 −l43 −1][2l + 1][2l′ + 1]!1/2 0, 0, 1, 0; (m)+11E+q2(−l+p) [l11 −l23][l31 −l43 −1][2l + 1][2l′ + 1]!1/2|0, 0, 0, 1; (m)⟩,(m)4=c4−q−2(l′+p′+1) [l11 −l23][l33 −l31 + 1][2l + 1][2l′ + 1]!1/2 1, 0, 0, 0; (m)+11−31E−q2(l+p−l′−p′) [l13 −l11][l33 −l31 + 1][2l + 1][2l′ + 1]!1/2 0, 1, 0, 0; (m)−31E+ [l11 −l23][l31 −l43 −1][2l + 1][2l′ + 1]!1/2 0, 0, 1, 0; (m)+11E+q2(l+p+1) [l13 −l11][l31 −l43 −1][2l + 1][2l′ + 1]!1/2|0, 0, 0, 1; (m)⟩,(m)5=c5q−2 [l13 −l11][l13 −l11 −1][2l + 1][2l + 2]!1/2 1, 0, 1, 0; (m)+11+11−31E−q2(−l+p) [l13 −l11][l11 −l23 + 1][2l + 1][2l + 2]!1/2 1, 0, 0, 1; (m)+11−31E−q2(−l+p−1) [l13 −l11][l11 −l23 + 1][2l + 1][2l + 2]!1/2 0, 1, 1, 0; (m)+11−31E+q2(−2l+2p−1) [l11 −l23][l11 −l23 + 1][2l + 1][2l + 2]!1/2 0, 1, 0, 1; (m)−31E,(m)6=c6−q−2(q2 + q−2) [l13 −l11 −1][l11 −l23 + 1][2][2l][2l + 2]!1/2 1, 0, 1, 0; (m)+11+11−31E+q2(−l+p)([l11 −l23] −q4(l+1)[l13 −l11 −1])([2][2l][2l + 2])1/21, 0, 0, 1; (m)+11−31E+q2(−l+p−1)([l11 −l23] −q4(l+1)[l13 −l11 −1])([2][2l][2l + 2])1/20, 1, 1, 0; (m)+11−31E+(q2 + q−2)q2(2p+1) [l11 −l23][l13 −l11][2][2l][2l + 2]!1/2 0, 1, 0, 1; (m)−31E,(m)7=c7q−2 [l11 −l23][l11 −l23 + 1][2l][2l + 1]!1/2 1, 0, 1, 0; (m)+11+11−31E

+q2(l+p+1) [l11 −l23][l13 −l11 −1][2l][2l + 1]!1/2 1, 0, 0, 1; (m)+11−31E+q2(l+p) [l11 −l23][l13 −l11 −1][2l][2l + 1]!1/2 0, 1, 1, 0; (m)+11−31E+q2(2l+2p+1) [l13 −l11][l13 −l11 −1][2l][2l + 1]!1/2 0, 1, 0, 1; (m)−31E,(m)8=c8q2 [l33 −l31 + 1][l33 −l31 + 2][2l′ + 1][2l′ + 2]!1/2 0, 0, 1, 1; (m)+11E+q2(l′−p′) [l33 −l31 + 2][l31 −l43 −1][2l′ + 1][2l′ + 2]!1/2 1, 0, 0, 1; (m)+11−31E−q2(l′−p′+1) [l33 −l31 + 2][l31 −l43 −1][2l′ + 1][2l′ + 2]!1/2 0, 1, 1, 0; (m)+11−31E+q2(2l′−2p′+3) [l31 −l43 −2][l31 −l43 −1][2l′ + 1][2l′ + 2]!1/2 1, 1, 0, 0; (m)+11−31−31E,(m)9=c9−q2(q2 + q−2) [l33 −l31 + 1][l31 −l43 −1][2][2l′][2l′ + 2]!1/2 0, 0, 1, 1; (m)+11E−q2(l′−p′)([l31 −l43 −2] −q−4(l′+1)[l33 −l31 + 1])([2][2l′][2l′ + 2])1/21, 0, 0, 1; (m)+11−31E+q2(l′−p′+1)([l31 −l43 −2] −q−4(l′+1)[l33 −l31 + 1])([2][2l′][2l′ + 2])1/20, 1, 1, 0; (m)+11−31E+q2(−2p′+1)(q2 + q−2) [l31 −l43 −2][l33 −l31 + 2][2][2l′][2l′ + 2]!1/2 1, 1, 0, 0; (m)+11−31−31E,(m)10=c10q2 [l31 −l43 −2][l31 −l43 −1][2l′][2l′ + 1]!1/2 0, 0, 1, 1; (m)+11E−q−2(l′+p′+1) [l33 −l31 + 1][l31 −l43 −2][2l′][2l′ + 1]!1/2 1, 0, 0, 1; (m)+11−31E+q−2(l′+p′) [l33 −l31 + 1][l31 −l43 −2][2l′][2l′ + 1]!1/2 0, 1, 1, 0; (m)+11−31E+q−2(2l′+2p′−1) [l33 −l31 + 1][l33 −l31 + 2][2l′][2l′ + 1]!1/2 1, 1, 0, 0; (m)+11−31−31E,(m)11=c11 [l13 −l11 −1][l33 −l31 + 2][2l + 1][2l′ + 1]!1/2 1, 0, 1, 1; (m)+11+11−31E

−q2(−l+p+1) [l11 −l23 + 1][l33 −l31 + 2][2l + 1][2l′ + 1]!1/2 0, 1, 1, 1; (m)+11−31E−q2(l′−p′+1) [l13 −l11 −1][l31 −l43 −2][2l + 1][2l′ + 1]!1/2 1, 1, 1, 0; (m)+11+11−31−31E+q2(−l+p+l′−p′+2) [l11 −l23 + 1][l31 −l43 −2][2l + 1][2l′ + 1]!1/2 1, 1, 0, 1; (m)+11−31−31E,(m)12=c12− [l11 −l23 + 1][l33 −l31 + 2][2l + 1][2l′ + 1]!1/2 1, 0, 1, 1; (m)+11+11−31E−q2(l+p+2) [l13 −l11 −1][l33 −l31 + 2][2l + 1][2l′ + 1]!1/2 0, 1, 1, 1; (m)+11−31E+q2(l′−p′+1) [l11 −l23 + 1][l31 −l43 −2][2l + 1][2l′ + 1]!1/2 1, 1, 1, 0; (m)+11+11−31−31E+q2(l+p+l′−p′+3) [l13 −l11 −1][l31 −l43 −2][2l + 1][2l′ + 1]!1/2 1, 1, 0, 1; (m)+11−31−31E,(m)13=c13− [l13 −l11 −1][l31 −l43 −2][2l + 1][2l′ + 1]!1/2 1, 0, 1, 1; (m)+11+11−31E+q2(−l+p+1) [l11 −l23 + 1][l31 −l43 −2][2l + 1][2l′ + 1]!1/2 0, 1, 1, 1; (m)+11−31E−q−2(l′+p′) [l13 −l11 −1][l33 −l31 + 2][2l + 1][2l′ + 1]!1/2 1, 1, 1, 0; (m)+11+11−31−31E+q2(−l+p−l′−p′+1) [l11 −l23 + 1][l33 −l31 + 2][2l + 1][2l′ + 1]!1/2 1, 1, 0, 1; (m)+11−31−31E,(m)14=c14 [l11 −l23 + 1][l31 −l43 −2][2l + 1][2l′ + 1]!1/2 1, 0, 1, 1; (m)+11+11−31E+q2(l+p+2) [l13 −l11 −1][l31 −l43 −2][2l + 1][2l′ + 1]!1/2 0, 1, 1, 1; (m)+11−31E+q−2(l′+p′) [l11 −l23 + 1][l33 −l31 + 2][2l + 1][2l′ + 1]!1/2 1, 1, 1, 0; (m)+11+11−31−31E+q2(l+p−l′−p′+2) [l13 −l11 −1][l33 −l31 + 2][2l + 1][2l′ + 1]!1/2 1, 1, 0, 1; (m)+11−31−31E,(m)15=c15 |1, 1, 1, 1; (m)⟩,(4.33)where l =12(m13 −m23), p = m11 −12(m13 + m23), l′ =12(m33 −m43) and p′ =

m31 −12(m33 + m43). The latest formula (4.33), in fact, represents a way in whichthe reduced basis is expressed in terms of the induced basis and vas versa it is nota problem for us to find the invert relation between these bases (see the Appendix).Taking into account all results obtained above we have proved the following as-sertionProposition 1 : The Uq[gl(2/2)]-module W q is decomposed as a direct sum (4.22) ofsixteen Uq[gl(2/2)0]-fidirmods V qk , k = 0, 1, ..., 15, every one of which is characterizedby a highest weight [m]k given in (4.32) and is spanned by a GZ basis (m)k givenin (4.33).The decomposition (4.22) of W q([m]) ≡W q([m13, m23, m33, m43]) can be re-written in the form 21W q([m])=V q(00)([m13, m23, m33, m43])min(1,2l)Mi=0min(1,2l′)Mj=0V q(10)([m13 −i, m23 + i −1, m33 −j + 1, m43 + j])min(2,2l)Mi=0V q(11)([m13 −i, m23 + i −2, m33 + 1, m43 + 1])min(2,2l′)Mj=0V q(20)([m13 −1, m23 −1, m33 −j + 2, m43 + j])min(1,2l)Mi=0min(1,2l′)Mj=0V q(21)([m13 −i −1, m23 + i −2, m33 −j + 2, m43 + j + 1])MV q(22)([m13 −2, m23 −2, m33 + 2, m43 + 2])(4.34)where V q(ab)([m](ab)) is an alternative notation of that submodule V qk ([m]k) with[m]k = [m](ab):V q(00)([m](00))≡V q(00)([m]) = V q0 ([m]),V q(10)([m](10))≡V q(10)([m]k) = V qk ([m]k),1 ≤k ≤4,

V q(11)([m](11))≡V q(11)([m]k) = V qk ([m]k),5 ≤k ≤7,V q(20)([m](20))≡V q(20)([m]k) = V qk ([m]k),8 ≤k ≤10,V q(21)([m](21))≡V q(21)([m]k) = V qk ([m]k),11 ≤k ≤14,V q(22)([m](22))≡V q(22)([m]15) = V q15([m]15). (4.35)In a natural way we denote by (m)(ab) the GZ basis of V q(ab)([m](ab)).

Thus, the basisof W q([m]) is spanned on the set of all possible patterns 21(m)(ab) ≡m13m23m33m43m12m22m32m42m110m310(ab),a, b ∈{0, 1, 2}(4.36)such thatm12=m13 −r −θ(a −2) −θ(b −2) + 1,m22=m23 + r −θ(a −1) −θ(b −1) −1,m32=m33 + a −s + 1,m42=m43 + b + s −1,(4.37)wherer=1, ..., 1 + min(a −b, 2l′),s=1, ..., 1 + min(⟨a⟩+ ⟨b⟩, 2l),(4.38)θ(x) =1if x ≥0,0if x < 0(4.39)and⟨i⟩=1for odd i,0for even i . (4.40)4.2.

Typical representations

The Uq[gl(2/2)]-module W q constructed is either irreducible or indecomposable.We can verify thatProposition 2 : The induced Uq[gl(2/2)]-module W q is irreducible if and only if thefollowing condition holds[l13 + l33 + 3][l13 + l43 + 3][l23 + l33 + 3][l23 + l43 + 3] ̸= 0. (4.41)In this case we say that the module W q is typical, otherwise it is called atypical.The proof of the latter proposition follows the one for the classical case consideredin Ref.

20 (cf. Ref.

18). Indeed, by the same argument we can conclude that W q isirreducible if and only ifE24E14E23E13E31E32E41E42 ⊗(M) ̸= 0.

(4.42)The latest condition (4.42) in turn can be proved, after some elementary calculations,to be equivalent to[E11 + E33 + 1][E11 + E44][E22 + E33][E22 + E44 −1](M) ̸= 0,(4.42′)which is nothing but the condition (4.41).Since Uq[gl(2/2)] is generated by the even generators and the odd Chevalleygenerators E23 and E32, any its representations in some basis is completely definedby the actions of these generators on the same basis. In the case of the typicalrepresentations the matrix elements of the generators in the reduced basis (4.33)can be obtained by keeping the conditions (4.1) and (4.41) valid and using therelations (3.1-5).

For the even generators we readily haveE11(m)k=(l11 + 1)(m)k,E22(m)k=(l12 + l22 −l11 + 2)(m)k,E12(m)k=([l12 −l11][l11 −l22])1/2 (m)+11k,E21(m)k=([l12 −l11 + 1][l11 −l22 −1])1/2 (m)−11k,E33(m)k=(l31 + 1)(m)k,

E43(m)k=(l32 + l42 −l31 + 2)(m)k,E34(m)k=([l32 −l31][l31 −l42])1/2 (m)+31k,E43(m)k=([l32 −l31 + 1][l31 −l42 −1])1/2 (m)−31k. (4.43)As the computations on finding the matrix elements of E23 and E32 are toocumbersome, we shall write down here only the final results.If we assume theformal notation[x1]...[xi][y1]...[yj] := [|x1|]...[|xi|][|y1|]...[|yj|](4.44)the generator E23 acts on the basis vectors (m)(ab), i.e.

on (m)k, as followsE23( ˜m)(00)=0,E23( ˜m)(10)=−q−2[l3−r,3 + ls+2,3 + 3][l3−r,3 −l11][l5−s,3 −l31 + 1][l13 −l23][l33 −l43]1/2( ˜m)+i2−j2−31(00),E23( ˜m)(ab)=q−2min(2,b−r+3)Xi=max(1,b−r+2)min(4,b+s+2)Xj=max(3,b+s+1)(−1)(b−1)i+b(j+1)×[li3 + lj3 + ⟨s⟩−⟨r⟩+ 3]×[li2 −l11 + 1][l7−j,2 −l31 + 1][l12 −l22][l32 −l42]1/2 [li2 −l3−i,2 + ⟨r⟩−1][2 −⟨r⟩][li3 −l3−i,3 + (−1)r]b/2×[lj2 −l7−j,2 −⟨s⟩+ 1][2 −⟨s⟩][lj3 −l7−j,3 −(−1)s](1−b)/2( ˜m)+i2−j2−31(10),a + b = 2,E23( ˜m)(21)=−q−22Xi=14Xj=3X⟨s+j⟩≤k=0,1≤⟨r+i⟩(−1)(1−k)i+kj×[li3 + lj3 −(−1)k ⟨i + j + s + r⟩+ 3]×[li2 −l11 + 1][l7−j,2 −l31 + 1][l12 −l22][l32 −l42]1/2 [lr2 −li2 + 2k −2][2 −k][l13 −l23]⟨i+r⟩/2×[l5−s,2 −lj2 + 2k][1 + k][l33 −l43]⟨s+j+1⟩/2( ˜m)+i2−j2−31(1+k,1−k) ,E23( ˜m)(22)=q−22Xi=14Xj=3(−1)i+j[li3 + lj3 + 3]

×[li3 −l11 −1][l7−j,3 −l31 + 3][l13 −l23][l33 −l43]1/2( ˜m)+i2−j2−31(21),(4.45)while the generator E32 has the following matrix elementsE32( ˜m)(00)=−q22Xi=14Xj=3[li3 −l11][l7−j,3 −l31][l13 −l23][l33 −l43]1/2( ˜m)−i2+j2+31(10),E32( ˜m)(10)=q22Xi=14Xj=3X⟨r+i⟩≤k=0,1≤⟨s+j⟩(−1)(1−k)i+k(j+1)×[li2 −l11][l7−j,2 −l31][l12 −l22][l32 −l42]1/2 [l3−i,3 −li3 + 2k −1][1 + k][l13 −l23]⟨i+r+1⟩/2×[l7−j,3 −lj3 + 2k −1][2 −k][l33 −l43]⟨s+j⟩/2( ˜m)−i2+j2+31(2−k,k),E32( ˜m)(ab)=−q2min(2,r−b+1)Xi=max(1,r−b)min(4,6−b−s)Xj=max(3,5−b−s)(−1)bi+(1−b)j×[li2 −l11][l7−j,2 −l31][l12 −l22][l32 −l42]1/2 [li2 −l3−i,2 −⟨r⟩+ 1][2 −⟨r⟩][li3 −l3−i,3 −(−1)r]b/2×[lj2 −l7−j,2 + ⟨s⟩−1][2 −⟨s⟩][lj3 −l7−j,3 + (−1)s](1−b)/2( ˜m)−i2+j2+31(21),a + b = 2,E32( ˜m)(21)=−q2(−1)r+s[lr3 −l11 −1][ls+2,3 −l31 + 2][l13 −l23][l33 −l43]1/2( ˜m)−r2+j2+31(22),j = 5 −s,E32( ˜m)(22)=0. (4.46)where ( ˜m)(ab) are obtained from (m)(ab) by rescaling( ˜m)k = 1ck(m)k,k = 0, 1, ..., 15(4.47).

5. ConclusionIn this work we have constructed all typical representations of the quantum su-peralgebras Uq[gl(2/2)] at the generic q leaving the coefficients ck (see (4.30-31))as free parameters.

The latter can be fixed by some additional conditions , for ex-ample the hermiticity condition. The quantum typical Uq[gl(2/2)]-module W q([m])

obtained has the same structure as the classical gl(2/2)-module (the module Win Ref.20) and can be decomposed into sixteen Uq[gl(2/2)0]-fidirmods V qk ([m]k),k = 0, 1, ..., 15. In general, the method used here is similar to that one of Ref.20.It is not difficult for us to see that forck = 1,k = 0, 1, .., 15,we obtain at q = 1 the classical results given in Refs.

20 (see also Ref. 21).

However,unlike the latter our approach in this paper avoids the use of the Clebsch-Gordancoefficients 33 which are not always known for higher rank (quantum and classical)algebras. We hope that the present approach can be applied for larger quantumsuperalgebras.

Following the classical programme 20,21 we can construct atypicalrepresentations of Uq[gl(2/2)] at generic q 22. Moreover, an extension of the presentinvestigations on the case when q is a root of unity is also possible 23.AcknowledgementsI am grateful to Prof. Tch.

Palev for numerous discussions. It is a pleasure forme to thank Prof. E. Celeghini and Dr. M. Tarlini for the kind hospitality at theFlorence University where the present investigations were reported.

I am thankfulto Profs. A. Barut, R. Floreanini and V. Rittenberg for useful discussions.I would like to thank Prof.Abdus Salam, the International Atomic EnergyAgency and UNESCO for the kind hospitality at the High Energy- and the Mathe-matical Section of the International Centre for Theoretical Physics, Trieste, Italy.The present work also was partially supported by the National Scientific Founda-tion of the Bulgarian Ministry of Science and Higher Education under the contractF-33.

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AppendixThe induced basis (4.20) is expressed in terms of the reduced basis through thefollowing invert relation|1, 0, 0, 0; (m)⟩=1c1q2(l+p+1) [l13 −l11 + 1][l31 −l43][2l + 1][2l′ + 1]!1/2(m)−11+311−1c2q2(−l+p) [l11 −l23 −1][l31 −l43][2l + 1][2l′ + 1]!1/2(m)−11+312+ 1c3q2(l+p+1) [l13 −l11 + 1][l33 −l31][2l + 1][2l′ + 1]!1/2(m)−11+313−1c4q2(−l+p) [l11 −l23 −1][l33 −l31][2l + 1][2l′ + 1]!1/2(m)−11+314,|0, 1, 0, 0; (m)⟩=−1c1q2 [l11 −l23][l31 −l43][2l + 1][2l′ + 1]!1/2(m)+311−1c2q2 [l13 −l11][l31 −l43][2l + 1][2l′ + 1]!1/2(m)+312−1c3q2 [l11 −l23][l33 −l31][2l + 1][2l′ + 1]!1/2(m)+313−1c4q2 [l13 −l11][l33 −l31][2l + 1][2l′ + 1]!1/2(m)+314,|0, 0, 1, 0; (m)⟩=1c1q2(l+p−l′−p′) [l13 −l11 + 1][l33 −l31 + 1][2l + 1][2l′ + 1]!1/2(m)−111−1c2q2(−l+p−l′−p′−1) [l11 −l23 −1][l33 −l31 + 1][2l + 1][2l′ + 1]!1/2(m)−112−1c3q2(l+p+l′−p′+1) [l13 −l11 + 1][l31 −l43 −1][2l + 1][2l′ + 1]!1/2(m)−113+ 1c4q2(−l+p+l′−p′) [l11 −l23 −1][l31 −l43 −1][2l + 1][2l′ + 1]!1/2(m)−114,|0, 0, 0, 1; (m)⟩=−1c1q−2(l′+p′) [l11 −l23][l33 −l31 + 1][2l + 1][2l′ + 1]!1/2(m)1−1c2q−2(l′+p′) [l13 −l11][l33 −l31 + 1][2l + 1][2l′ + 1]!1/2(m)2

+ 1c3q2(l′−p′+1) [l11 −l23][l31 −l43 −1][2l + 1][2l′ + 1]!1/2(m)3+ 1c4q2(l′−p′+1) [l13 −l11][l31 −l43 −1][2l + 1][2l′ + 1]!1/2(m)4,|1, 0, 1, 0; (m)⟩=1c5q2(2l+2p+1) [l13 −l11 + 1][l13 −l11 + 2][2l + 1][2l + 2]!1/2(m)−11−11+315−1c6q2(2p−1) [2][l13 −l11 + 1][l11 −l23 −1][2l][2l + 2]!1/2(m)−11−11+316+ 1c7q2(−2l+2p−1) [l11 −l23 −2][l11 −l23 −1][2l][2l + 1]!1/2(m)−11−11+317,|1, 0, 0, 1; (m)⟩=−1c5q2(l+p+2) [l13 −l11 + 1][l11 −l23][2l + 1][2l + 2]!1/2(m)−11+315−1c6[2]1/2q2(p+1)(q2l+2[l13 −l11] −q−2l−2[l11 −l23 −1])(q2 + q−2) ([2l][2l + 2])1/2(m)−11+316+ 1c7q2(−l+p+1) [l13 −l11][l11 −l23 −1][2l][2l + 1]!1/2(m)−11+317+ 1c8q−2(l′+p′) [l33 −l31 + 1][l31 −l43][2l′ + 1][2l′ + 2]!1/2(m)−11+318−1c9[2]1/2q2(−p′+1)(q2l′+2[l31 −l43 −1] −q−2l′−2[l33 −l31])(q2 + q−2) ([2l′][2l′ + 2])1/2(m)−11+319−1c10q2(l′−p′+1) [l33 −l31][l31 −l43 −1][2l′][2l′ + 1]!1/2(m)−11+3110,|0, 1, 1, 0; (m)⟩=−1c5q2(l+p+1) [l13 −l11 + 1][l11 −l23][2l + 1][2l + 2]!1/2(m)−11+315−1c6[2]1/2q2p(q2l+2[l13 −l11] −q−2l−2[l11 −l23 −1])(q2 + q−2) ([2l][2l + 2])1/2(m)−11+316+ 1c7q2(−l+p) [l13 −l11][l11 −l23 −1][2l][2l + 1]!1/2(m)−11+317−1c8q2(−l′−p′+1) [l33 −l31 + 1][l31 −l43][2l′ + 1][2l′ + 2]!1/2(m)−11+318+ 1c9[2]1/2q2(−p′+2)(q2l′+2[l31 −l43 −1] −q−2l′−2[l33 −l31])(q2 + q−2) ([2l′][2l′ + 2])1/2(m)−11+319

+ 1c10q2(l′−p′+2) [l33 −l31][l31 −l43 −1][2l′][2l′ + 1]!1/2(m)−11+3110,|0, 1, 0, 1; (m)⟩=1c5q2 [l11 −l23][l11 −l23 + 1][2l + 1][2l + 2]!1/2(m)+315+ 1c6q2 [2][l11 −l23][l13 −l11][2l][2l + 2]!1/2(m)+316+ 1c7q2 [l13 −l11 −1][l13 −l11][2l][2l + 1]!1/2(m)+317,|1, 1, 0, 0; (m)⟩=1c8q2 [l31 −l43][l31 −l43 + 1][2l′ + 1][2l′ + 2]!1/2(m)−11+31+318+ 1c9q2 [2][l31 −l43][l33 −l31][2l′][2l′ + 2]!1/2(m)−11+31+319+ 1c10q2 [l33 −l31 −1][l33 −l31][2l′][2l′ + 1]!1/2(m)−11+31+3110,|0, 0, 1, 1; (m)⟩=1c8q−2(2l′+2p′+1) [l33 −l31 + 1][l33 −l31 + 2][2l′ + 1][2l′ + 2]!1/2(m)−118−1c9q2(−2p′+1) [2][l33 −l31 + 1][l31 −l43 −1][2l′][2l′ + 2]!1/2(m)−119+ 1c10q2(2l′−2p′+1) [l31 −l43 −2][l31 −l43 −1][2l′][2l′ + 1]!1/2(m)−1110,|1, 1, 1, 0; (m)⟩=−1c11q2(l+p+1) [l13 −l11 + 1][l31 −l43][2l + 1][2l′ + 1]!1/2(m)−11−11+31+311+ 1c12q2(−l+p) [l11 −l23 −1][l31 −l43][2l + 1][2l′ + 1]!1/2(m)−11−11+31+312−1c13q2(l+p+1) [l13 −l11 + 1][l33 −l31][2l + 1][2l′ + 1]!1/2(m)−11−11+31+313+ 1c14q2(−l+p) [l11 −l23 −1][l33 −l31][2l + 1][2l′ + 1]!1/2(m)−11−11+31+3114,|1, 1, 0, 1; (m)⟩=1c11q2 [l11 −l23][l31 −l43][2l + 1][2l′ + 1]!1/2(m)−11+31+3111

+ 1c12q2 [l13 −l11][l31 −l43][2l + 1][2l′ + 1]!1/2(m)−11+31+3112+ 1c13q2 [l11 −l23][l33 −l31][2l + 1][2l′ + 1]!1/2(m)−11+31+313+ 1c14q2 [l13 −l11][l33 −l31][2l + 1][2l′ + 1]!1/2(m)−11+31+3114,|1, 0, 1, 1; (m)⟩=1c11q2(l+p−l′−p′) [l13 −l11 + 1][l33 −l31 + 1][2l + 1][2l′ + 1]!1/2(m)−11−11+3111−1c12q2(−l+p−l′−p′−1) [l11 −l23 −1][l33 −l31 + 1][2l + 1][2l′ + 1]!1/2(m)−11−11+3112−1c13q2(l+p+l′−p′+1) [l13 −l11 + 1][l31 −l43 −1][2l + 1][2l′ + 1]!1/2(m)−11−11+3113+ 1c14q2(−l+p+l′−p′) [l11 −l23 −1][l31 −l43 −1][2l + 1][2l′ + 1]!1/2(m)−11−11+3114,|0, 1, 1, 1; (m)⟩=−1c11q−2(l′+p′) [l11 −l23][l33 −l31 + 1][2l + 1][2l′ + 1]!1/2(m)−11+3111−1c12q−2(l′+p′) [l13 −l11][l33 −l31 + 1][2l + 1][2l′ + 1]!1/2(m)−11+3112+ 1c13q2(l′−p′+1) [l11 −l23][l31 −l43 −1][2l + 1][2l′ + 1]!1/2(m)−11+3113+ 1c14q2(l′−p′+1) [l13 −l11][l31 −l43 −1][2l + 1][2l′ + 1]!1/2(m)−11+3114,|1, 1, 1, 1; (m)⟩=1c15(m)−11−11+31+31.

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