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Unitarity and Complete Reducibility of Certain

arXiv:hep-th/9303096v5 2 Jul 1993UQMATH-93-02hep-th/9303096Unitarity and Complete Reducibility of CertainModules over Quantized Affine Lie AlgebrasY.-Z.Zhang and M.D.GouldDepartment of Mathematics, University of Queensland, Brisbane, Qld 4072, AustraliaAbstract:Let Uq( ˆG) denote the quantized affine Lie algebra and Uq(G(1)) the quantized nontwisted affineLie algebra. Let Ofin be the category defined in section 3.

We show that when the deformationparameter q is not a root of unit all integrable representations of Uq( ˆG) in the category Ofin arecompletely reducible and that every integrable irreducible highest weight module over Uq(G(1))corresponding to q > 0 is equivalent to a unitary module.

1IntroductionQuantum (super)groups, or more precisely quantum universal enveloping (super)algebras or forshort quantized (super)algebras, are defined as q deformations of classical universal envelopingalgebras of finite-dimensional simple Lie (super)algebras [1][2][3]. The definition for the quan-tized finite-dimensional simple Lie algebras can be extended to infinite-dimensional affine Liealgebras, or even to arbitrary Kac-Moody algebras, with symmetrizable, generalized Cartanmatrices in the sense of Kac[4].

In this paper we shall be concerned with the case of quantizedaffine Lie algebras.Quantized affine Lie algebras and their representations are important in, among others, theso-called Yang-Baxterization method for obtaining spectral parameter dependent solutions tothe quantum Yang-Baxter equation [5][6] and the q-deformed WZNW CFT’s[7][8]. It is known[9][10] that most algebras and representations of interest in physics and mathematics have corre-sponding q deformations.

In particular, for quantized simple Lie algebras, all finite-dimensionalrepresentations are known to be completely reducible[9]. However, for quantized affine Lie al-gebras, only some isolated results are avaiable.In particular, the complete reducibility andunitarity of representations have not been clarified.

In fact the latter remains unproved even forthe quantized finite-dimensional simple Lie algebra case.In this paper we will address the problems of complete reducibility and unitarity of certainrepresentations for quantized affine Lie algebras. Our main results are the proofs of completereducibility and unitarity of some important modules over the quantized affine Lie algebras.The paper is set up in the following way.After recalling, in section 2, some basic factson quantized affine Lie algebras, in section 3 we investigate representations of the quantizedaffine Lie algebras.

The main result of this section is theorem 3.1 which says that all integrablerepresentations of the quantized affine Lie algebras with weight spectrum bounded from aboveare completely reducible. In section 4 we prove the unitarity of every integrable irreduciblehighest weight module over quantized nontwisted affine Lie algebras; our main results are statedin theorem 4.2, 4.4 and corollary 4.3.

We conclude, in section 5, with a brief discussion of ourmain results.2PreliminariesWe start with the definition of the quantum affine Lie algebra Uq( ˆG). Let A0 = (aij)1≤i,j≤r bea symmetrizable Cartan matrix.Let G stand for the finite-dimensional simple Lie algebra withsymmetric Cartan matrix A0sym = (asymij) = (αi, αj),i, j = 1, 2, ..., r, where r is the rank ofG.

Let A = (aij)0≤i,j≤r be a symmetrizable, generalized Cartan matrix in the sense of Kac.Let ˆG denote the affine Lie algebra associated with the corresponding symmetric Cartan matrixAsym = (asymij) = (αi, αj),i, j = 0, 1, ...r. The quantum algebra Uq( ˆG) is defined to be a Hopf1

algebra with generators: {ei, fi, qhi (i = 0, 1, ..., r), qd} and relations,qh.qh′ = qh+h′(h, h′ = hi (i = 0, 1, ..., r), d)qheiq−h = q(h,αi)ei ,qhfiq−h = q−(h,αi)fi[ei, fj] = δijqhi −q−hiq −q−11−aijXk=0(−1)ke(1−aij−k)ieje(k)i= 0 (i ̸= j)1−aijXk=0(−1)kf (1−aij−k)ifjf (k)i= 0 (i ̸= j)(1)wheree(k)i=eki[k]q!,f (k)i= f ki[k]q! ,[k]q = qk −q−kq −q−1(2)The algebra Uq( ˆG) is a Hopf algebra with coproduct, counit and antipode similar to the caseof Uq(G):∆(qh) = qh ⊗qh ,h = hi, d∆(ei) = q−hi/2 ⊗ei + ei ⊗qhi/2∆(fi) = q−hi/2 ⊗fi + fi ⊗qhi/2S(a) = −qhρaq−hρ ,a = ei, fi, hi, d(3)where ρ is the half-sum of the positive roots.

We have omitted the formula for counit since wedo not need them.For quasitriangular Hopf algebras, there exists a distinguished element [1][11]u =XiS(bi)ai(4)where ai and bi are coordinates of the universal R-matrix R = Pi ai ⊗bi. One can show that uhas inverseu−1 =XiS−2(bi)ai(5)and satisfiesS2(a) = uau−1 ,∀a ∈Uq( ˆG)∆(u) = (u ⊗u)(RT R)−1(6)where RT = T(R), T is the twist map: T(a ⊗b) = b ⊗a , ∀a ∈Uq( ˆG).Proposition 2.1: Ω= uq−2hρ belongs to the center of Uq( ˆG), i.e.

it is a Casimir operator.We may equivalently work with the coproduct ¯∆and antipode ¯S defined by¯∆(qh) = qh ⊗qh ,h = hi, d2

¯∆(ei) = qhi/2 ⊗ei + ei ⊗q−hi/2¯∆(fi) = qhi/2 ⊗fi + fi ⊗q−hi/2¯S(a) = −q−hρaqhρ ,a = ei, fi, hi, d(7)Corresponding to the coproduct and antipode (7) we have another form of the R-matrix, denotedas ¯R. If we write ¯R = Pi ¯ai ⊗¯bi, then we have¯u =Xi¯S(¯bi) ¯ai ,¯u−1 =Xi¯S−2(¯bi) ¯ai(8)which satisfy¯S2(a) = ¯ua¯u−1 ,∀a ∈Uq( ˆG)¯∆(¯u) = (¯u ⊗¯u)( ¯RT ¯R)−1(9)Proposition 2.2: ¯Ω= ¯u−1q−2hρ is the Casimir operator of Uq( ˆG) with coproduct and antipodegiven by (7).Proposition 2.3: The Casimir operators Ωand ¯Ωhave the properties: Ω= S(Ω) , ¯Ω= ¯S(¯Ω).We define a conjugate operation † and an anti-involution θ on Uq( ˆG) byd† = d ,h†i = hi ,e†i = fi ,f †i = ei ,i = 0, 1, ...rθ(qh) = q−h ,θ(ei) = fi ,θ(fi) = ei ,θ(q) = q−1(10)which extend uniquely to an algebra anti-automorphism and anti-involution on all of Uq( ˆG),respectively, so that (ab)† = b†a† ,θ(ab) = θ(b)θ(a) ,∀a, b ∈Uq( ˆG).Throughout the paper, we assume that q is not a root of unit and use the notations:(n)q = 1 −qn1 −q ,[n]q = qn −q−nq −q−1 ,qα = q(α,α)expq(x) =Xn≥0xn(n)q!

,(n)q! = (n)q(n −1)q ... (1)q(adqxα)xβ = [xα , xβ]q = xαxβ −q(α , β)xβxα(11)3Complete ReducibilityRepresentations of Uq( ˆG) are known to be isomorphic (as a linear space) to corresponding repsof U( ˆG) [10].

LetUq( ˆG) = Uq(N−) ⊗Uq(H) ⊗Uq(N+)(12)be the triangular decomposition of Uq( ˆG), generated by the fi’s, qh (h ∈H) and ei’s, respectively.Let V denote an Uq( ˆG)-module. We say that V is H-diagonalizable ifV =Mλ∈H∗Vλ(13)3

where, Vλ = {v ∈V |qhv = qλ(h)v , h ∈H} be the weight spaces corresponding to the weight λ.Following Kac[4], we introduce the notationsΠ(V ) = {λ ∈H∗|Vλ ̸= 0} ,D(λ) = {µ ∈H∗|µ ≤λ , λ ∈H∗}(14)and consider the category Ofin defined byDefinition 3.1: Ofin is the category of Uq( ˆG) modules V which are H-diagonalizable with finitedimensional weight spaces and in which there exist a finite number of elements λ1 , ... , λs ∈H∗such thatΠ(V ) ⊂∪si=1 D(λi)(15)(15) implies that weights in Ofin are bounded from above.Example: Highest weight modules.We say that an Uq( ˆG)-module V is a highest weight module with the highest weight Λ if thereexists a non-zero vector v ∈V such thatei(v) = 0(i = 0, 1, ..., r) ,qhv = qΛ(h)v (h ∈H)V = Uq( ˆG)v(16)The vector v is called as highest weight vector. We deduce from (12) and (16),V =Mλ≤ΛVλ ,VΛ = Cv ,dim Vλ < ∞(17)which implies that a highest weight module lies in Ofin.Proposition 3.1 (Lusztig [10]): For any Λ ∈H∗, there exists a unique, up to isomorphism,irreducible highest weight Uq( ˆG)-module L(Λ) with highest weight Λ.Following Kac[4], we have the followingDefinition 3.2: Let V be an Uq( ˆG)-module.

A vector v ∈Vλ is called primitive of weight λ ifthere exists a submodule U in V such thatv ̸∈U ;eiv ∈U(18)The weight λ is called a primitive weight.Example: If V is an irreducible Uq( ˆG)-module, so that the only proper submodule is U = {0},then a weight vector is primitive implies that v ̸= 0 and eiv = 0.We now state two propositions (3.2, 3.3, below) and one lemma (3.1, below) analogousto Kac’s classical results (compare Kac[4], proposition 9.3, lemma 9.5 and proposition 9.9,respectively).Proposition 3.2: Let V be a non-zero Uq( ˆG)-module from Ofin.a) V contains a non-zero weight vector v such that ei(v) = 0; in particular V contains a primitive4

vector.b) The following conditions are equivalent:(i) V is irreducible. (ii) V is a highest weight Uq( ˆG)-module and any primitive vector of V is a highest weight vector.

(iii) V ≃L(Λ) for some Λ ∈H∗.The condition (iii) means that the L(Λ) exhaust all irreducible modules from Ofin.Lemma 3.1: Let V be an Uq( ˆG)-module from Ofin. If for any two primitive weights λ and µ ofV , the inequality λ ≥µ implies λ = µ, then the module V is completely reducible.Proof: The proposition (3.2) and lemma (3.1) are proved exactly as in Kac ([4], proposition9.3 and lemma 9.5).✷Let Q = Pri=0 Zαi denote the root lattice and set Q+ = Pri=0 Z+αi.Proposition 3.3: a) Let V (Λ) be an Uq( ˆG)-module with highest weight Λ.

If 2(Λ+ρ, β) ̸= (β, β)for every β ∈Q+, β ̸= 0, then V (Λ) is irreducible.b) Let V be an Uq( ˆG)-module from Ofin. If for any two primitive weights λ and µ of V , suchthat λ −µ = β > 0, one has 2(λ + ρ, β) ̸= (β, β), then V is completely reducible.Proof: We mimic Kac’s proof in the classical case.

To this end, we first prove the followingLemma 3.2: Let V be an Uq( ˆG)-module.a) If there exists v ∈V such that eiv = 0 for all i = 0, 1, ..., r and qhv = qΛ(h)v for some Λ ∈H∗and all h ∈H, thenΩv = q−(Λ,Λ+2ρ)v(19)b) If, furthermore, V = Uq( ˆG)v, thenΩ|V = q−(Λ,Λ+2ρ)IV(20)Proof: One can show that the universal R-matrix R of Uq( ˆG) can be written in the form[7]R = I ⊗I +Xta′t ⊗b′t!· qPri=1 Hi⊗Hi+c⊗d+d⊗c(21)where {a′t} and {b′t} are the basis of the subalgebras of Uq( ˆG) generated by {eiq−hi/2} and{qhi/2fi}, i = 0, 1, ..., r, respectively; c = h0 + hψ ,ψ is the highest root of G; {Hi} and{Hi} (i = 1, 2, ..., r) satisfyrXi=1Λ(Hi)Λ′(Hi) = (Λ0, Λ′0) ,∀Λ = (Λ0, κ, σ) , Λ′ = (Λ′0, κ′, σ′) ∈H∗(22)So the Casimir Ωin proposition 2.1 takes the formΩ= q−Pri=1 HiHi−dc−cd−2hρ +XtS(b′t) q−Pri=1 HiHi−dc−cd · a′t q−2hρ(23)5

Acting on v, only the first term survives,Ωv = q−Pri=1 HiHi−dc−cd−2hρv = q−(Λ,Λ+2ρ)v(24)where use has been made of (22) and (Λ, Λ′) = (Λ0, Λ′0) + κσ′ + σκ′. This proves a).

Part b)follows from (24) and proposition 2.1.✷Corollary 3.1: a) If V is a highest weight Uq( ˆG) module with highest weight Λ, thenΩ= q−(|Λ+ρ|2−|ρ|2)IV(25)b) If V is an Uq( ˆG)-module from Ofin and v is a primitive vector with weight λ, then there existsa submodule U ⊂V such that v ̸∈U andΩv = q−(|Λ+ρ|2−|ρ|2)v (mod U)(26)Now we are in the position to prove proposition 3.3. Assume that V (Λ) is reducible.

Thenproposition 3.2b) implies that there exists a primitive weight λ = Λ −β, where β > 0 and thusfrom corollary 3.1a) we haveq−(Λ,Λ+2ρ) = q−(Λ−β,Λ−β+2ρ)(27)which gives 2(Λ + ρ, β) = (β, β) since q is not a root of unity. This leads to a contradiction andthus we prove a).We now prove b).

We may assume that the Uq( ˆG)-module is indecomposable. Then, locally,the Casimir operator Ωhas the same spectrum on V .

We thus obtain from corollary 3.1b)q−(|λ+ρ|2−|ρ|2) = q−(|µ+ρ|2−|ρ|2)(28)for any two primitive weights λ and µ. Since q is not a root of unity, the above equation gives|λ+ρ|2 = |µ+ρ|2 for any two primitive weights λ and µ.

Therefore, we must have λ = µ. Indeed,if this is not the case, then we deduce 2(λ + ρ, β) = (β, β), which contradicts the condition ofthe proposition. Now point b) follows from lemma 3.1.✷Definition 3.3: An Uq( ˆG)-module V is called integrable if V is H-diagonalizable and if ei andfi (i = 0, 1, ..., r) are locally nilpotent endomorphisms of V .Let Π(Λ) denote the set of weights of the Uq( ˆG)-module L(Λ) and D+ = {λ ∈H∗|(λ, αi) ≥0, 0 ≤i ≤r} the set of dominant integral weights.Proposition 3.5: a) Let V be an Uq( ˆG)-module from Ofin and λ be a primitive weight.

If V isintegrable, then λ ∈D+.b) The highest weight Uq( ˆG)-module L(Λ) with highest weight Λ is integrable iffΛ ∈D+.Proof: Part a) is proved following the same arguments as in Lusztig ([10], proposition 3.2) andpart b) follows from [9][10].✷6

We now state our main result (complete reducibility theorem) in this section.Theorem 3.1: Every integrable Uq( ˆG)-module V from Ofin is completely reducible, that is, isisomorphic to a direct sum of modules L(Λ), Λ ∈D+.Proof: We check that if λ and µ are primitive weights such that λ −µ = β, where β ∈Q+/{0},then2(λ + ρ, β) ̸= (β, β)(29)This can easily be done as follows. By means of proposition 3.5a) and the fact that (ρ, β) > 0for all β ∈Q+/{0}, we have2(λ + ρ, β) −(β, β) = (λ + (λ −µ) + 2ρ, β) = (λ + µ + 2ρ, β) > 0(30)The theorem then follows from proposition 3.3b).✷4UnitarityIn this section we will focus our attention on quantized nontwisted affine Lie algebras Uq(G(1)).Analogous conclusions are true for the twisted case.

We first introduce the followingDefinition 4.0: An Uq(G(1))-module V is called unitary if V can be equipped with an innerproduct < | > such that, for all a ∈Uq(G(1))< a†v|w >=< v|aw > ,∀v, w ∈V(31)Equivalently, if π is the representation of Uq(G(1)) afforded by V , then V is called unitaryprovidedπ(a†) = π(a)† ,∀a ∈Uq(G(1))(32)where † on the r.h.s. denotes Hermitian conjugate.Lemma 4.0: Every integrable highest weight Uq(G(1))-module L(Λ) carries a unique, up to aconstant factor, and well-defined nondegenerate inner product <|>.

With respect to thisinner product, L(Λ) decomposes into an orthogonal direct sum of weight spaces.Proof: This can be easily proved following the similar arguments as in Kac ([4], proposition9.4).✷Proposition 4.0 (Kac): Let Λ ∈D+ and λ ∈Π(Λ). Then |Λ + ρ|2 −|λ + ρ|2 ≥0 and equalityholds iffλ = Λ.4.1.

Let ˆG = sl(2)(1). Fix a normal ordering in the positive root system ∆+ of sl(2)(1):α, α + δ, ..., α + nδ, ..., δ, 2δ, ..., mδ, ... , ... , β + lδ, ... , β(33)7

where α and β are simple roots and l, m, n ≥0; δ = α + β is the minimal positive imaginaryroot. Let us introduce standard generatorsEα = eαq−hα/2 ,Eβ = eβq−hβ/2Fα = qhα/2fα ,Fβ = qhβ/2fβ(34)then,S(E†α) = −Fα ,S(E†β) = −FβS(F †α) = −Eα ,S(F †β) = −Eβ(35)Construct Cartan-Weyl generators Eγ , Fγ = θ(Eγ) ,γ ∈∆+ of Uq(sl(2)(1)) as follows[12]: Wedefine˜Eδ = [(α, α)]−1q [Eα, Eβ]qEα+nδ = (−1)n ad ˜Eδn EαEβ+nδ =ad ˜Eδn Eβ , ...˜Enδ = (α, α)]−1q [Eα+(n−1)δ, Eβ]q(36)where [ ˜Enδ, ˜Emδ] = 0 for any n, m > 0.

For any n > 0 there exists a unique element Enδ [12]which satisfies [Enδ , Emδ] = 0 for any n, m > 0 and the relation˜Enδ =Xk1p1 + ... + kmpm = n0 < k1 < ... < kmq(α,α) −q−(α,α)Pi pi−1p1! ...

pm!(Ek1δ)p1... (Ekmδ)pm(37)Then the vectors Eγ and Fγ = θ(Eγ), γ ∈∆+ defined above are the Cartan-Weyl generators forUq(sl(2)(1)).

Moreover,Theorem 4.1 (Khoroshkin-Tolstoy[12]): The universal R-matrix for Uq(sl(2)(1)) may be writtenasR=Πn≥0expqα((q −q−1)(Eα+nδ ⊗Fα+nδ))·exp Xn>0n[n]−1qα (qα −q−1α )(Enδ ⊗Fnδ)!·Πn≥0expqα((q −q−1)(Eβ+nδ ⊗Fβ+nδ))· q12hα⊗hα+c⊗d+d⊗c(38)where c = hα + hβ. The order in the product (38) concides with the chosen normal order (33).We haveLemma 4.1:S(E†α+nδ) = −qn(α,β)Fα+nδ ,S(E†β+nδ) = −qn(α,β)Fβ+nδ8

S(F †α+nδ) = −q−n(α,β)Eα+nδ ,S(F †β+nδ) = −q−n(α,β)Eβ+nδS( ˜E†nδ) = −qn(α,β) ˜Fnδ ,S(E†nδ) = −qn(α,β)FnδS( ˜F †nδ) = −q−n(α,β) ˜Enδ ,S(F †nδ) = −q−n(α,β)Enδ(39)Proof: The proof follows, from (35), (36) and (37) and similar relations for Fγ = θ(Eγ), byinduction in n.✷Corollary 4.1:S(Fα+nδ) = −q−n(α,β)S2(E†α+nδ) = −q−n(α,β)−(α+nδ,2ρ)E†α+nδS(Fβ+nδ) = −q−n(α,β)S2(E†β+nδ) = −q−n(α,β)−(β+nδ,2ρ)E†β+nδS(Fnδ) = −q−n(α,β)S2(E†nδ) = −q−n(α,β)−(nδ,2ρ)E†nδ(40)We are now ready to stateTheorem 4.2: Every integrable highest weight module L(Λ) over Uq(sl(2)(1)) corresponding toq > 0 is equivalent to a unitary module.Proof: In the limit q →1, the Uq(sl(2)(1))-module L(Λ) reduces to the corresponding moduleof U(sl(2)(1)) [10] and thus is equivalent to a unitary module according to Kac[4]. We now showthat for 0 < q < 1 and q > 1 the module L(Λ) is equivalent to a unitary module.

By lemma4.0, one only need to show that if < | > is a nondegenerate inner product on L(Λ) such that< v|v >> 0 for a highest weight vector v, then the restriction of < | > to L(Λ)λ is positivedefinite for each weight λ in L(Λ). We prove this by induction on ht(Λ −λ) (the height of(Λ −λ)).

Let λ ∈Π(Λ)/{Λ}.(i). For 0 < q < 1:we use the Casimir Ω= uq−2hρ.

We have, from the R-matrix (38),u=X{l,n,k}Al,n,k(q) S(Fβ)k0 ... S(Fβ+Mδ)kM ... ... S(FLδ)nL ... S(Fδ)n1· ... S(Fα+Nδ)lN ... S(Fα)l0q−12 hαhα−cd−dc(Eα)l0 ... (Eα+Nδ)lN ...·(Eδ)n1 ... (ELδ)nL ... ... (Eβ+Mδ)kM ... (Eβ)k0(41)where {l} = {l0, l1, ..., lN, ...},{n} = {n1, n2, ..., nL, ...} , {k} = {k0, k1, ..., kM, ...};the con-stants Al,n,k(q) are given byAl,n,k=(q −q−1)l0+l1+...+lN+...(l0)qα! ... (lN)qα!

...(q −q−1)k0+k1+...+kM+...(k0)qα! ... (kM)qα!

...·1n1 ... LnL ... (q −q−1)n1+n2+...+nL+...[1]n1qα ... [L]nLqα ... n1! ... nL!

...(42)and satisfy(−1)l0+...+lN+...(−1)n1+...+nL+...(−1)k0+...+kM+...Al,n,k(q) > 0for 0 < q < 1(43)9

Then the corollary 4.1 implies thatΩ=q−12 hαhα−cd−dc−2hρ +′X{l,n,k}ˆAl,n,k(q) (E†β)k0 ... (E†β+Mδ)kM ... ...·(E†Lδ)nL ... (E†δ)n1 ... (E†α+Nδ)lN ... (E†α)l0q−12hαhα−cd−dc(Eα)l0 ... (Eα+Nδ)lN· ... (Eδ)n1 ... (ELδ)nL ... ... (Eβ+Mδ)kM ... (Eβ)k0q−2hρ(44)where ˆAl,n,k(q) = (−1)l0+...+lN+...(−1)n1+...+nL+...(−1)k0+...+kM+...Al,n,k(q) and so by (43)ˆAl,n,k(q) > 0for0 < q < 1(45)and P′ denotes the sum over all {l, n, k} ̸= {0, 0, 0}.Computing < v|Ω|v > in two different ways, we obtainq−(|Λ+ρ|2−|ρ|2) −q−(|λ+ρ|2−|ρ|2)< v|v >=′X{l,n,k}ˆAl,n,k(q) q−2(λ,ρ)· < (Eα)l0 ... (Eα+Nδ)lN ...· (Eδ)n1 ... (ELδ)nL ... ... (Eβ+Mδ)kM ... (Eβ)k0v|q−12 hαhα−cd−dc· (Eα)l0 ... (Eα+Nδ)lN ... (Eδ)n1 ... (ELδ)nL ... ... (Eβ+Mδ)kM ... (Eβ)k0v >(46)By the inductive assumption the r.h.s. of (46) is non-negative thanks to eq.(45).

Using proposi-tion 4.0 we deduce that < v|v >≥0 for 0 < q < 1. Since < | > is non-degenerate on L(Λ)λwe conclude that for 0 < q < 1 it is positive definite on L(Λ).(ii).

For q > 1:In this case we work with the coproduct and antipode (7).Let us introduce the standardgenerators¯Eα = eαqhα/2 ,¯Eβ = eβqhβ/2¯Fα = q−hα/2fα ,¯Fβ = q−hβ/2fβ(47)then¯S( ¯E†α) = −¯Fα ,¯S( ¯E†β) = −¯Fβ¯S( ¯F †α) = −¯Eα ,¯S( ¯F †β) = −¯Eβ(48)and we haveLemma 4.2:¯S( ¯Fα+nδ) = −qn(α,β)+(α+nδ,2ρ) ¯E†α+nδ¯S( ¯Fβ+nδ) = −qn(α,β)+(β+nδ,2ρ) ¯E†β+nδ¯S( ¯Fnδ) = −qn(α,β)+(nδ,2ρ) ¯E†nδ(49)Proof: Similar to the proof of lemma 4.1.✷10

Now we define inductively,˜¯Eδ = [(α, α)]−1q [ ¯Eα, ¯Eβ]q−1¯Eα+nδ = (−1)n ad ˜¯Eδn ¯Eα¯Eβ+nδ =ad ˜¯Eδn ¯Eβ , ...˜¯Enδ = [(α, α)]−1q [ ¯Eα+(n−1)δ, ¯Eβ]q−1˜¯Enδ =Xk1p1 + ... + kmpm = n0 < k1 < ... < kmq−(α,α) −q(α,α)Pi pi−1p1! ...

pm! ( ¯Ek1δ)p1...( ¯Ekmδ)pm(50)and similarly for ¯Fγ = θ( ¯Eγ).

Then we immediately obtain, from the R-matrix (38), our matrix( ¯RT)−1 :( ¯RT)−1=X{l,n,k}Al,n,k(q−1) ( ¯Eα)l0 ... ( ¯Eα+Nδ)lN ... ( ¯Eδ)n1 ... ( ¯ELδ)nL ...· ... ( ¯Eβ+Mδ)kM ... ( ¯Eβ)k0 ⊗( ¯Fα)l0 ... ( ¯Fα+Nδ)lN ... ( ¯Fδ)n1 ... ( ¯FLδ)nL· ... ... ( ¯Fβ+Mδ)kM ... ( ¯Fβ)k0 · q−12 hα⊗hα−c⊗d−d⊗c(51)where the constants Al,n,k(q−1) satisfy(−1)l0+...+lN+...(−1)n1+...+nL+...(−1)k0+...+kM+...Al,n,k(q−1) > 0 ,forq > 1(52)We deduce, from (51),¯R=(I ⊗¯S) ¯R−1 =X{l,n,k}Al,n,k(q−1) ( ¯Fα)l0 ... ( ¯Fα+Nδ)lN ... ( ¯Fδ)n1 ... ( ¯FLδ)nL ...· ... ( ¯Fβ+Mδ)kM ... ( ¯Fβ)k0 ⊗q12hα⊗hα+c⊗d+d⊗c · ¯S( ¯Eβ)k0 ... ¯S( ¯Eβ+Mδ)kM ...· ... ¯S( ¯ELδ)nL ... ¯S( ¯Eδ)n1 ... ¯S( ¯Eα+Nδ)lN ... ¯S( ¯Eα)l0(53)Thus we obtain the following Casimir operator¯Ω=¯S(¯Ω) = ¯S(¯u−1q−2hρ) = q2hρ ¯S(¯u−1)= q2hρX{l,n,k}Al,n,k(q−1) ¯S( ¯Fβ)k0 ... ¯S( ¯Fβ+Mδ)kM ... ...· ¯S( ¯FLδ)nL ... ¯S( ¯Fδ)n1 ... ¯S( ¯Fα+Nδ)lN ... ¯S( ¯Fα)l0 · q12hαhα+cd+dc·( ¯Eα)l0 ... ( ¯Eα+Nδ)lN ... ( ¯Eδ)n1 ... ( ¯ELδ)nL ... ... ( ¯Eβ+Mδ)kM ... ( ¯Eβ)k0(54)which, using the lemma 4.2, takes the form¯Ω=q12hαhα+cd+dc+2hρ +′X{l,n,k}ˆAl,n,k(q−1) ( ¯E†β)k0 ... ( ¯E†β+Mδ)kM ... ...·( ¯E†Lδ)nL ... ( ¯E†δ)n1 ... ( ¯E†α+Nδ)lN ... ( ¯E†α)l0q12 hαhα+cd+dc( ¯Eα)l0 ... ( ¯Eα+Nδ)lN· ... ( ¯Eδ)n1 ... ( ¯ELδ)nL ... ... ( ¯Eβ+Mδ)kM ... ( ¯Eβ)k0(55)11

where ˆAl,n,k(q−1) = (−1)l0+...+lN+...(−1)n1+...+nL+...(−1)k0+...+kM+...Al,n,k(q−1) and so by (52)ˆAl,n,k(q−1) > 0forq > 1(56)Computing < v|¯Ω|v > in two different ways as above, we obtainq|Λ+ρ|2−|ρ|2 −q|λ+ρ|2−|ρ|2< v|v >=′X{l,n,k}ˆAl,n,k(q−1) q2(λ,ρ)· < ( ¯Eα)l0 ... ( ¯Eα+Nδ)lN ...· ( ¯Eδ)n1 ... ( ¯ELδ)nL ... ... ( ¯Eβ+Mδ)kM ... ( ¯Eβ)k0v|q12hαhα+cd+dc· ( ¯Eα)l0 ... ( ¯Eα+Nδ)lN ... ( ¯Eδ)n1 ... ( ¯ELδ)nL ... ... ( ¯Eβ+Mδ)kM ... ( ¯Eβ)k0v >(57)By the inductive hypothesis we have that the r.h.s. of (57) is non-negative for q > 1 thanks tothe formula (56).

Therefore, we deduce from proposition 4.0 that < v|v >≥0 for q > 1.Then the non-degeneracy of < | > on L(Λ) implies that < v|v >> 0 for q > 1.✷4.2. General case:ˆG = G(1).

Fix some order in the positive root system ∆+ of G(1), whichsatisfies an additional condition,α + nδ ≤kδ ≤(δ −β) + lδ(58)where α , β ∈∆0+ , ∆0+ is the positive root system of G ; k , l , n ≥0 and δ is the minimalpositive imaginary root.Let us as before introduce standard generators,Ei = eiq−hi/2 ,Fi = qhi/2fi ,i = 0, 1, ..., r(59)then we haveS(E†i ) = −Fi ,S(F †i ) = −Ei(60)Cartan-Weyl generators Eγ and Fγ = θ(Eγ) ,γ ∈∆+ may be constructed inductively asfollows[12]. We start from the simple roots.

If γ = α + β ,α < γ < β, is a root and there areno other positive roots α′ and β′ between α and β such that γ = α′ + β′, then we setEγ = [Eα , Eβ]q = EαEβ −q(α,β)EβEα(61)When we get the root δ, we use the following formula for roots γ + nδ and roots (δ −γ) + nδ,for γ ∈∆0+,˜E(i)δ= [(αi, αi)]−1q [Eαi, Eδ−αi]q ,Eαi+nδ = (−1)n ad ˜E(i)δn Eαi ,Eδ−αi+nδ =ad ˜E(i)δn Eδ−αi ,...,˜E(i)nδ = [(αi, αi)]−1q [Eαi+(n−1)δ, Eδ−αi]q(62)12

Then we repeat the above inductive proceduce to obtain other real root vectors Eγ+nδ , Eδ−γ+nδ ,γ ∈∆0+.Finally, the imaginary root vectors E(i)nδ are defined through ˜E(i)nδ by the relation(37) with α there changing to αi.Then, the above operators E(i)nδ ,F (i)nδ = θ(E(i)nδ)(i =1, 2, ..., r) , Eγ ,Fγ = θ(Eγ) are the Cartan-Weyl generators of Uq(G(1)). MoreoverTheorem 4.3 (Khoroshkin-Tolstoy[12]): The universal R-matrix Uq(G(1)) may be written inthe following form,R= Πγ∈∆re+ , γ<δ expqγ q −q−1Cγ(q) Eγ ⊗Fγ!

!·expXn>0rXi,j=1Cnij(q)(q −q−1)(E(i)nδ ⊗F (j)nδ )· Πγ∈∆re+ , γ>δ expqγ q −q−1Cγ(q) Eγ ⊗Fγ! !· qPri,j=1 (a−1sym)ijhi⊗hj+c⊗d+d⊗c(63)where c = h0 +hψ, ψ is the highest root of G; (Cnij(q)) = (Cnji(q)) , i, j = 1, 2, ..., r, is the inverseof the matrix (Bnij(q)) ,i, j = 1, 2, ..., r withBnij(q) = (−1)n(1−δij)n−1 qnij −q−nijqj −q−1jq −q−1qi −q−1i,qij = q(αi,αj) ,qi ≡qαi(64)and Cγ(q) is a normalizing constant defined by[Eγ , Fγ] = Cγ(q)q −q−1qhγ −q−hγ,γ ∈∆re+(65)The order in the product of the R-matrix concides with the chosen normal ordering (58) in ∆+.We now state an importantRemark: Cγ(q) have the following general propertyCγ(q) = Cγ(q−1) > 0for q > 0 ,q ̸= 1(66)as shown in our previous paper[13].We have the followingLemma 4.3: For any α ∈∆0+,S(E†α) = −q(α,α−2ρ)/2Fα ,S(F †α) = −q−(α,α−2ρ)/2Eα(67)Proof: We prove them by induction.

The results obviously are valid for α = αi , i = 1, 2, ..., r,a simple root since we have (αi, αi −2ρ) = 0. Now we show that the results are also true forEα+β = [Eα , Eβ]q and Fα+β = [Fβ , Fα]q−1.

We haveS(E†α+β) = S(E†α)S(E†β) −q(α,β)S(E†β)S(E†α)(68)13

which by the inductive assumption givesS(E†α+β)=q(α,α−2ρ)/2+(β,β−2ρ)/2 FαFβ −q(α,β)FβFα=q(α,β)+(α,α−2ρ)/2+(β,β−2ρ)/2 FβFα −q−(α,β)FαFβ=−q(α+β,α+β−2ρ)/2Fα+β(69)Similarly, we haveS(F †α+β) = −q−(α+β,α+β−2ρ)/2Eα+β(70)This completes our proof.✷Corollary 4.2: For any α ∈∆0+,S(E†δ−α) = −q(δ−α,δ−α−2ρ)/2Fδ−α ,S(F †δ−α) = −q−(δ−α,δ−α−2ρ)/2Eδ−α(71)Proof: The results are true for α = ψ, the highest root. Then the results follow, from lemma4.3 and Eδ−α = [Eβ , Eδ−(α+β)]q and Fδ−α = [Fδ−(α+β) , Fβ]q−1, by induction as above.✷Lemma 4.4: For any α ∈∆0+,S(E†α+nδ) = −q(α,α−2ρ)/2−n(δ,ρ)Fα+nδS(F †α+nδ) = −q−(α,α−2ρ)/2+n(δ,ρ)Eα+nδS(E†δ−α+nδ) = −q(δ−α,δ−α−2ρ)/2−n(δ,ρ)Fδ−α+nδS(F †δ−α+nδ) = −q−(δ−α,δ−α−2ρ)/2+n(δ,ρ)Eδ−α+nδS( ˜E(i)†nδ ) = −q−n(δ,ρ) ˜F (i)nδ .S( ˜F (i)†nδ ) = −qn(δ,ρ) ˜E(i)nδS(E(i)†nδ ) = −q−n(δ,ρ)F (i)nδ ,S(F (i)†nδ ) = −qn(δ,ρ)E(i)nδ(72)Proof: The proof follows, from (60), (62) and lemma 4.3 and corollary 4.2, by induction inn.✷Our main result is:Theorem 4.4: Every integrable highest weight module L(Λ) over Uq(G(1)) corresponding toq > 0 is equivalent to a unitary module.Proof: The proof is similar to the one of theorem 4.2 for Uq(sl(2)(1)) case thanks to lemma 4.3,corollary 4.2, lemma 4.4 and remark (66).✷Corollary 4.3: Every integrable highest weight module L(Λ) over Uq(G) corresponding to q > 0is equivalent to a unitary module.5Concluding RemarksTo summarize, in this paper we have investigated the complete reducibility and unitarity ofcertain modules over the quantized affine Lie algebras.

In our proofs the Casimir operator (thus14

the universal R-matrix) plays a key role and our approach is actually a modest imitation ofKac’s one[4] and the one in [14] used in proving the similar results for the classical affine Liealgebras and finite-dimensional simple Lie superalgebras, respectively.The complete reducibility theorem 3.1 implies that the tensor product L(Λ) ⊗L(Λ′) of theintegrable irreducible highest weight modules L(Λ) and L(Λ′) is completely reducible and theirreducible components are integrable highest weight representations. This makes possible thecomputation of link polynomials [15][16] associated with the quantized affine Lie algebras.

Thepoint is that the universal R-matrix for the quantized affine Lie algebras can be shown to satisfythe conjugation rule, R† = RT ; therefore, braid generators are diagonalizable[14] on L(Λ)⊗L(Λ′),regardless of multiplicity. Our results also allow us to construct generalized Gelfand invariants[17] of the quantized affine Lie algebras.

All details will be reported in a separate publication.Acknowledgements:Y.Z.Z. would like to thank Anthony John Bracken for contineous encouragement and suggestions,to thank Loriano Bonora for communication of preprint [12] and to thank M.Scheunert for manypatient explanations on quantum groups during July and August of last year.

The financialsupport from Australian Research Council is gratefully acknowledged.15

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