Q-Operator and Fusion Relations for U q ( C ( 2 ) ( 2 ))

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Q-Operator and Fusion Relations for U _q ( C ⁽ ² ⁾ ( 2 ))

IVAN CHI-HO IP¹ and ANTON M. ZEITLIN²^,³^,⁴

1Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan. e-mail: ivan.ip@ipmu.jp URL: http://member.ipmu.jp/ivan.ip

2Department of Mathematics, Columbia University, Room 509, MC 4406, 2990 Broadway, New York, NY 10027, USA. e-mail: zeitlin@math.columbia.edu

URL: http://math.columbia.edu/∼zeitlin

3Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany 4IPME RAS, V.O. Bolshoj pr., 61, 199178 St. Petersburg, Russia.

URL: http://www.ipme.ru/zam.html

Received: 17 January 2014 / Revised: 16 April 2014 / Accepted: 17 April 2014

Published online: 6 May 2014 – © Springer Science+Business Media Dordrecht 2014

Abstract. The construction of the Q-operator for twisted affine superalgebraU_q(C⁽²⁾(2))is given. It is shown that the corresponding prefundamental representations give rise to evaluation modules some of which do not have a classical limit, which nevertheless appear to be a necessary part of fusion relations.

Mathematics Subject Classification (2010). Primary 17B37, 81R50.

Keywords. Q-operator, integrable model, fusion relations, twisted affine superalgebra, R-matrix.

1. Introduction

The Q-operator and its generalizations are important ingredients in the study of quantum integrable models. Namely, eigenvalues of the transfer matrices, corresponding to various representations, can be expressed in terms of eigenvalues of the Q-operator, which has less-complicated analytic properties. These features of the Q-operators were first noticed by Baxter the early 70s in the case of ver- tex models [1]. Later, after the quantum group interpretation of the quantum integrable models it was realized that the original Baxter Q-operator corresponds to the integrable model based on the simplest nontrivial quantum affine algebra Uq(sl(2)). A natural question was to generalize this notion to the higher rank and give a proper representation-theoretic meaning to these fundamental building blocks for transfer matrix eigenvalues. The first idea in that direction was given in the papers of Bazhanov et al. [3,4] in the context of the construction of integrable structure of conformal field theory: the interpretation of Q-operators for Uq(sl(2)) as transfer matrices for certain infinite-dimensional representations of the Borel subalgebra of Uq(sl(2)). Later their results were generalized in [2,13]

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to the case of Uq(sl(n)). Finally, in the recent preprint of Frenkel and Hernan- dez [6] the full representation-theoretic description of Q-operators was given for large class of integrable models based on any untwisted quantum affine algebra Uq(g) and connected to the earlier description of the transfer matrix eigenvalues via the q-characters [7]. The infinite-dimensional representations corresponding to the Q-operator, which the authors called “prefundamental representations” were constructed just before that in [9].

At the same time, some analogs of the Q-operators were constructed in this way in the case of superalgebras [5,15,17]. In this article we improve the constructions of [15,16]. In that paper an attempt to construct the Q-operator and asso- ciated fusion relation for transfer matrices was made in the case of Uq(C⁽²⁾(2))≡ Uq(sl⁽²⁾(2|1)). However, the construction given there leads to only partial result:

half of the resulting transfer matrices were built “by hands” out of Q-operators and did not seem to correspond to any finite-dimensional representation of Uq(C⁽²⁾(2)). In this paper we solve this ambiguity, by allowing some representa- tions to have no classical limit (q→1). The approach we are using allows to show explicitly the similarity betweenUq(A⁽¹⁾₁ )andUq(C⁽²⁾(2))previously noticed on the level of universal R-matrices [12].

The structure of the article is as follows. In Section 2 we fix the notations and describe the relation between finite-dimensional representations of Uq(osp(2|1)) and Uq(sl(2)), previously noticed on the level of modular double [10]. The approach, which can be generalized to higher rank superalgebras, is that we find representations ofUq(osp(2|1))inside the tensor product of finite-dimensional representation of U₋i q(sl(2)) and two-dimensional Clifford algebra. Such representation splits into two irreducible representations which differ by the parity of the highest weight and have equal dimensions. It is notable that the even-dimensional irreducible representations obtained in this way do not have the classical limit.

We also give explicit formulas for R-matrix in these representations. In Section 3 we consider evaluation modules forUq(C⁽²⁾(2)), which can be obtained in a sim- ilar fashion from evaluation modules of U₋i q(A⁽¹⁾₁ ). We explicitly find the resulting trigonometric R-matrix and its matrix coefficients (with the details of calcula- tions in the Appendix). We also introduce in Section 3 the prefundamental representations forUq(C⁽²⁾(2)) and study in detail the relations in the Grothendieck ring of prefundamental representations combined with evaluation representations.

The relations in the Grothendieck ring lead to relations between transfer matrices and Q-operators: in Section4 we correct the constructions of [15], where the integrable structure of superconformal field theory was studied, now changing “fusion- like” relations by the true fusion relations.

2. Quantum Superalgebra Uq(osp(2|1)) and its Representations

We define the quantum superalgebraUq(osp(2|1)) as follows. It is a Hopf algebra generated by even element K and odd elements E and F such that

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{E,F} :=EF+FE=K−K⁻¹ q+q⁻¹, KE =q²EK,

KF =q⁻²FK,

where the corresponding co-product is (E)=E⊗K+1⊗E,

(F)=F⊗1+K⁻¹⊗F, (K)=K⊗K.

Let us choose the (odd) Clifford generators ξ, η satisfying

ξ²=η²=1, ξη= −ηξ (1)

which acts in the space C¹^|¹:=span{|+,|−}, where |−, |+ are odd and even vectors correspondingly, by

π(ξ)= 0 1

1 0

, π(η)=

0 i

−i 0

, (2)

such that π(iξη)=

1 0 0 −1

. (3)

The following notation will play a crucial role in relating the superalgebra and the classical case via the spinor representation:

DEFINITION 2.1. We denote by

q_∗:= −i q (4)

and writing q:=e^π^{i b}²,q_∗:=e^π^{i b}²^∗, we have b²=b²_∗+1

2. (5)

Then we have the following proposition observed in [10], which can be proved by direct computation.

PROPOSITION 2.2. If E,F,K generateUq_∗(sl(2)), then

E=ξE, F=ηF, K=iξηK (6)

generate Uq(osp(2|1)).

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Therefore, we are now able to relate the representations of Uq_∗(sl(2)) and Uq(osp(2|1)). Let us do it explicitly.

Consider thes+1=2l+1-dimensional representationVs ofUq_∗(sl(2))with basis e^l_m, m= −l, . . . ,l

and action

K·e^l_m=q_∗^2me^l_m, H·e^l_m=(2m)e^l_m,

E·e^l_m= [l−m]q_∗e^l_m₊₁, F·e^l_m= [l+m]q_∗e^l_m₋₁,

where formally K=q_∗^H and [n]q:=^q_qⁿ₋⁻_q^q₋⁻1ⁿ is the quantum number.

The generators E,F,K naturally act on Vs⊗C¹^|¹ by means of the Uq_∗(sl(2)) action, and it decomposes as

Ws=Vs⊗C¹^|¹=W_s⁺⊕W_s⁻, (7)

where W_s^± has highest weight w^±s =e^l_l⊗ |± and spanned by

W_s^±=span{w^±s ,F·w^±s ,F²·w^±s , . . . ,F^s·ws^±}. (8) Let e_m^l_,±:=e^l_m⊗ |± be the natural basis of Vs⊗C^1|1. Note that e^l_m_,− is an odd vector while e^l_m_,+ is even. Then the action of E,F,K can be written explicitly as follows:

PROPOSITION 2.3.

K·e^l_m_,±= ±q_∗^2me^l_m_,±,

= ±i⁻^2mq^2me_m^l_,±, E·e^l_m_,±= [l−m]q_∗e_m^l₊₁_,∓,

=i^l⁻^m⁻¹{l−m}qe^l_m₊₁_,∓, F·e^l_m_,±= ∓i[l+m]q_∗e^l_m_−1,∓

= ∓i^l⁺^m{l+m}qe^l_m₋₁_,∓

where {n}q:=^q⁻ⁿ_q⁻⁽⁻¹⁾₊_q₋1ⁿ^qⁿ =i¹⁻ⁿ[n]q_∗.

We notice that the representations of even dimension is something which we do not encounter in the classical case, namely all the finite-dimensional irreducible representations of Lie superalgebra osp(2|1)are odd-dimensional.

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Remark 2.4. It is well known that the finite-dimensional irreducible representations of osp(2|1) Lie superalgebra have odd dimension only (see e.g. [8]). One can relate W_s^± for evens to those by considering classical limit. Due to our normalization, to do that one has to proceed through the following steps. First, one has to rescale e^l_m so that F e_m^l =e^l_m₋₁ and renormalize E so that E=^q_q⁺₋^q_q⁻₋¹1E. ThenE=ξE, and F are such that the commutation relations on W_s^± in the limit q−→1 are such that [E,F] =H, i.e. the commutation relations of osp(2|1). Such limiting procedure is not possible in the case of even-dimensional W_s^± as the coefficients will not con- verge.

EXAMPLE 2.5. For l = ¹₂, the representation on (W₁^±, π1) with basis {e¹₁^/_/²₂_,±, e^1/2_−1/2,∓}is given by

π1(K)=

∓i q 0 0 ∓i q⁻¹

, π1(H)=

1 0 0 −1

, π1(E)=

0 1

0 0

, π1(F)=

0 0

∓i 0

.

For l=1, the representation on (W₂^±, π2) with basis {e₁¹_,±,e¹₀_,∓,e¹₋₁_,±}is given by

π2(K)=

⎛

⎝∓q² 0 0

0 ∓1 0

0 0 ∓q⁻²

⎞

⎠, π2(H)=

⎛

⎝2 0 0

0 0 0

0 0 −2

⎞

⎠,

π2(E)=

⎛

⎝0 1 0 0 0 i(q⁻¹−q)

0 0 0

⎞

⎠, π2(F)=

⎛

⎝ 0 0 0

±(q⁻¹−q) 0 0

0 ±i 0

⎞

⎠.

Now we will find the formula for the R-matrix acting in tensor product of W_s^±. Let

exp_q(x)=

∞ n=0

xⁿ

nq! (9)

where nq=¹⁻₁₋^q_qⁿ. The the following Theorem holds.

THEOREM 2.6. The universal R matrix is given by R=QR

where Q:=Cq

H⊗H

∗ 2 with C:=¹₂(1⊗1+iξη⊗1+1⊗iξη+ξη⊗ξη) such that C· |(−1)¹ ⊗ |(−1)² =(−1)¹²|(−1)¹ ⊗ |(−1)², i∈ {0,1}, (10)

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and

R:=exp_q₋2

∗ (i(q_∗⁻¹−q_∗)E⊗F)

=exp₋_q−2(−(q+q⁻¹)E⊗F)

= anEⁿ⊗Fⁿ (11)

where

an=(−1)ⁿq¹²ⁿ⁽ⁿ⁻¹⁾(q+q⁻¹)ⁿ

{n}q! (12)

The proof is given in Appendix.

Finally, let us give for completeness the explicit matrix coefficients of R. Namely, we find the pairing for R_l₁_,_l₂=R|_W_s^±

1⊗W_s^±₂

e^l_m¹

1,1⊗e^l_m²

2,2,Rl1,l2(e^l_m¹

1,1⊗e^l_m²

2,2)

wherei∈{0,1}indicates the parity, namely|±=|(−1). Let us fixl1,l2 and write e^l_m_, for e^l_m_,±.

PROPOSITION 2.7.

e^l_m¹

1,1⊗e^l_m²

2,2,R_l₁_,_l₂(e^l_m¹

1,1⊗e^l_m²

2,2)

=0 if m₁−m₁=m₂−m₂ or m₁−m₁=m₂−m₂<0.

Otherwise let n=m₁−m1, we have

e^l_m¹

1,1⊗e^l_m²

2,2,R_l₁_,_l₂(e^lm¹₁,1⊗e^l_m²₂_,₂)

=i⁽^l¹⁻^m¹⁺^l²⁺^m²⁻¹⁾ⁿ^−2m¹^m²(−1)¹²⁺ⁿq¹²ⁿ⁽ⁿ^−1)+2m¹^m²·

·(q+q⁻¹)ⁿ {n}q!

{l₁−m₁}q! {l1−m1−n}q!

{l₂+m₂}q! {l2+m2−n}q!

In terms of q_∗ and using the standard[n]q_∗ instead, we get

=q

1

2n(n−1)+2m₁m₂

∗ (−1)¹²(q_∗−q_∗⁻¹)ⁿ [n]q_∗!

[l1−m1]q_∗! [l1−m1−n]q_∗!

[l2+m2]q_∗! [l2+m2−n]q_∗! Note that there are no more i’s using the q_∗ notation.

EXAMPLE 2.8. ForW₁⁺⊗W₁⁺, let the basis be {e¹₁^/_/²₂_,+,e¹₋^/₁²_/₂_,−}⊗{e¹₁^/_/²₂_,+,e¹₋^/₁²_/₂_,−}. Then R is given by

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R¹

2,¹2 =

⎛

⎜⎜

⎜⎝ q

1

∗2 0 0 0

0 q⁻

1

∗2 (1−q_∗⁻²)q_∗¹² 0

0 0 q⁻

1

∗ 2 0

0 0 0 −q_∗¹²

⎞

⎟⎟

⎟⎠

EXAMPLE 2.9. For W₂⁺⊗W₂⁺, let the basis be {e¹_1,+,e_0,−¹ ,e_−1,+¹ } ⊗ {e_1,+¹ ,e¹_0,−, e¹₋₁_,+}. Then R is given by

R_1,1=

⎛

⎜⎜

⎝

q_∗² 0 0 0 0 0 0 0 0

0 1 0 q_∗²−q_∗⁻² 0 0 0 0 0

0 0 q_∗⁻² 0 q_∗⁻²(q_∗⁻¹−q_∗) 0 (q_∗²−q_∗⁻²)(1−q_∗⁻²) 0 0

0 0 0 1 0 0 0 0 0

0 0 0 0 −1 0 (q_∗²−q_∗⁻²)(q_∗+q_∗⁻¹) 0 0

0 0 0 0 0 1 0 q_∗²−q_∗⁻² 0

0 0 0 0 0 0 q_∗⁻² 0 0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 q_∗²

⎞

⎟⎟

⎠

Finally we give some remarks about the Casimir operator. In Uq_∗(sl(2)), it is known that the center is generated by the Casimir operator given by (up to some additive constant):

C_sl(2)=F E+q_∗K+q_∗⁻¹K⁻¹

(q_∗−q_∗⁻¹)² (13)

By Proposition 2.2, it is obvious that C_sl(₂₎ commutes with our generators. How- ever, it is not an element of Uq(osp(2|1)). Instead, the element

C_osp(2|1):=ηξC_sl(2)=FE−qK−q⁻¹K⁻¹

(q+q⁻¹)² (14)

will be an element inUq(osp(2|1)) super-commuting with the generators{E,F,K}. By construction, its square C_osp(2|1):= −C²_sl(2) is in the center of Uq(osp(2|1)), given by

C_osp(2|1)= −F²E²+ q−q⁻¹

(q+q⁻¹)²(q²K+q⁻²K⁻¹)FE+q²K²+q⁻²K⁻²

(q+q⁻¹)⁴ (15) up to an additive constant. Under a rescaling of the generators, this is precisely the Casimir element of Uq(osp(2|1)) found in [14].

Now it is also clear that the representations W_s^± correspond to the positive and negative spectrum of the square root of the Casimir element

C_osp(2|1)=ηξC_sl(2).

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3. Evaluation Modules forUq(C⁽²⁾(2)) and Prefundamental Representations The quantum affine superalgebra Uq(C⁽²⁾(2)) is generated by Ei,Fi,Ki, i=0,1, where Ei and Fi are odd, with Cartan matrix given by

A=

2 −2

−2 2

.

In particular, we have

KiEj=q^a^{i j}EjKi, KiFj=q⁻^a^{i j}FjKi, (16)

and in addition the Serre relations

E_i³Ej+ {3}qE_i²EjEi− {3}qEiEjE_i²−EjE_i³=0 (17) F_i³Fj+ {3}qF_i²FjFi− {3}qFiFjF_i²−FjF_i³=0 (18)

where {3}q=^q_q³₊⁺_q^q₋⁻1³. Furthermore, for later convenience we modify the scaling of Fi and use instead the following commutation relations:

{Ei,Fi} =Ki−K⁻¹_i

q+q⁻¹ . (19)

3.1. EVALUATION MODULES FORUq(C⁽²⁾(2))AND TRIGONOMETRICR-MATRIX

One check easily that we have the following spinor representation as in the Uq(osp(2|1)) case:

Ej=Ejξ, Fj=Fjη, Kj=iξηKj, (20)

and we also have the evaluation modules induced from(A⁽₁¹⁾)q_∗ given by E₁→λE, E₀→λF

F1→λ⁻¹F, F0→λ⁻¹E K1→K, K0→K⁻¹

Then using the 2-dimensional representation of the Clifford algebra, we can consider its action as before on Vs⊗C¹^|¹ and decompose it into Ws(λ):=Ws(λ)⁺⊗ Ws(λ)⁻.

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PROPOSITION 3.1. The action on the evaluation module Ws(λ)^± with basis e^l_m_,±, s=2l, m= −l, . . . ,l, is given by

E1·e^l_m_,±=λ[l−m]q_∗e^l_m₊₁_,∓

E0·e^l_m_,±=λ[l+m]q_∗e^l_m₋₁_,∓

F1·e^l_m_,±= ∓iλ⁻¹[l+m]q_∗e^l_m₋₁_,∓

F0·e^l_m_,±= ∓iλ⁻¹[l−m]q_∗e^l_m_+1,∓

K1·e^l_m_,±= ±q_∗^2me^l_m_,±

K0·e^l_m_,±= ±q_∗⁻^2me^l_m_,±

Kδ·e^l_m_,±=e_m^l_,±

In the case s=1, one can solve for the R matrix explicitly.

PROPOSITION 3.2. The R matrix for s=1, Ws(λ1)¹⊗Ws(λ2)², i∈ {+,−}, is, up to scalar, given by

R

⎛

⎜⎜

⎝

1−z²q_∗² 0 0 0

0 1q_∗(1−z²) 2z(1−q_∗²) 0 0 1z(1−q_∗²) 2q_∗(1−z²) 0

0 0 0 −12(1−z²q_∗²)

⎞

⎟⎟

⎠, (21)

where z=^λ_λ²₁. Alternatively, let λi=e^xⁱ, then we can cast it in trigonometric terms:

R

⎛

⎜⎜

⎜⎝

sinh(x₁−x₂−lnq_∗) 0 0 0

0 1sinh(x₁−x₂) 2sinh(lnq_∗) 0 0 1sinh(lnq_∗) 2sinh(x₁−x₂) 0

0 0 0 −₁₂sinh(x₁−x₂−lnq_∗)

⎞

⎟⎟

⎟⎠.

(22) In the general case, one has to calculate the action of the generators corresponding to the imaginary roots. The explicit calculation is given in the Appendix and the explicit form of the R-matrix is presented in Theorem A.4.

3.2. PREFUNDAMENTAL REPRESENTATIONS AND THE GROTHENDIECK RING

Let us consider the Verma modules corresponding to evaluation modules of Uq(C⁽²⁾(2)). Namely, let us start from the following representation ofUq(osp(2|1)): Ws^±= {F^k·w^±s }^∞_k₌₀, (23) where ws^±:=e^l_l⊗ |± as before such that K·w^±s = ±q_∗^sw^±s .

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Writing |k_±:=F^kw^±_s , the basis are related to e^l_m_,± of the s+1-dimensional module W_s^± from before by

|0±=ws^±=e^l_l_,±,

|k_±=F^k·ws^±=F^k·e^l_l_,±=i⁻^k [2l]q_∗!

[2l−k]q_∗!e^l_l₋_k_,±(−₁₎k. (24) Note that |k_± is an even vector when±(−1)^k= +1.

This gives rise to the following evaluation module of the upper Borel partb₊ of Uq(C⁽²⁾(2)) on Ws^±:

E0|k_±=λF|k_±=λ|k+1_±,

E1|k_±=λE|k_±=λ[k]q_∗[s−k+1]q_∗|k−1_±,

K0|k_±=K⁻¹|k_±= ±q^2kq_∗⁻^s|k_±= ±(−1)^kq_∗^2k⁻^s|k_±, K1|k_±=K|k_±= ±q^−2kq_∗^s|k_±= ±(−1)^kq_∗^−2k⁺^s|k_±.

Furthermore, we see that whens∈Z_≥0, W_s^±(λ) has a block diagonal form such that in the Grothendieck ring of the representation ofb₊,

[Ws^±(λ)] = [W_s^±(λ)] + [W₋^±(−_s₋₂¹⁾^s⁺¹(λ)]. (25) Let us define the prefundamental (or q-oscillator) representation of b₊ of Uq(Cq⁽²⁾(2)). The q-oscillator algebra is generated byα₊, α₋,H such that

qα₊α₋+q⁻¹α₋α₊= − 1

q+q⁻¹, [H, α_±] = ±2α_±, (26) where α± are considered as odd elements. We consider the Fock modules

_±=span{α^k_±|0_±:H|0_±=0, α_∓|0_±=0}^∞_k₌₀, (27) where the vacuum vectors |0_± are even. Then, we have an important lemma.

LEMMA 3.3. The following substitution provides an infinite-dimensional representa- tion of b₊:

ρ_±(λ):E1=λα_±, E0=λα_∓, K1=q^±H, K0=q^∓H. (28) Let us consider the tensor productρ₊(λμ)⊗ρ₋(λμ⁻¹). The action ofEi is given by

E1=λ(μα+⊗q^−H+1⊗μ⁻¹α−)=:λ(a₋+b₋) E0=λ(μα₋⊗q^H+1⊗μ⁻¹α₊)=:λ(a₊+b₊)

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so that we have the commutation relations qa₋a₊+q⁻¹a₊a₋= − μ²

q+q⁻¹, qb₊b₋+q⁻¹b₋b₊= − μ⁻²

q+q⁻¹,

a_δ₁b_δ₂= −q^2δ¹^δ²b_δ₂a_δ₁, δi∈ {±}, or, in q_∗ notation we have

q_∗a₋a₊−q_∗⁻¹a₊a₋= μ² q_∗−q_∗⁻¹, q_∗b₊b₋−q_∗⁻¹b₋b₊= μ⁻²

q_∗−q_∗⁻¹, a_δ₁b_δ₂=q_∗^2δ¹^δ²b_δ₂a_δ₁,

which is similar to the bosonic case considered in [4]. Hence as in [4], the tensor product ρ+(λμ)⊗ρ−(λμ⁻¹) decomposes as

ρ₊(λμ)⊗ρ₋(λμ⁻¹)=^∞

m=0

ρ⁽^m⁾, (29)

where

ρ⁽^m⁾: |ρ_k⁽^m⁾ =(a₊+b₊)^k(a₊−γb₊)^m|0+⊗ |0−, (30) for k∈Z≥0 and γ= −q_∗²ⁿ,n∈Zany constant. Note that |ρ_k⁽^m⁾ is even when k+m is even.

Let μ=q

s 2+¹₂

∗ . Then the action ofb₊ is given by

ρ₊(λμ)⊗ρ₋(λμ⁻¹)(E0)|ρ_k⁽^m⁾ =λ|ρ_k⁽^m₊⁾₁,

ρ₊(λμ)⊗ρ₋(λμ⁻¹)(E1)|ρ_k⁽^m⁾ =λ[k]q_∗[s−k+1]q_∗|ρ_k⁽^m₋⁾₁ +c⁽_k^m⁾|ρ⁽_k^m⁻¹⁾, where c⁽_k^m⁾ are constants not necessary in what follows.

We observe that the representation of b₊ has a block diagonal form defined by ρ⁽^m⁾, which resembles the Verma module Ws^± with a shift in the factors of Ki. Hence in the Grothendieck ring of representation of b₊ we obtain

[ρ₊(λμ)⊗ρ₋(λμ⁻¹)] = ^∞

m=0

[U₋s−2m⊗Ws⁽⁻¹⁾^m(λ)] (31)

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where Up is the 1-dimensional representation such that E1,E0 act trivially as 0, while K1,K0 acts as q_∗^p,q_∗⁻^p respectively. Indeed, the action of K1 on U₋s−2m⊗ Ws⁽⁻¹⁾^m(λ) is given by multiplication by

(q_∗⁻^s⁻^2m)·((−1)^m(−1)^kq_∗⁻^2k⁺^s)=(−1)^k⁺^mq_∗⁻²⁽^k⁺^m⁾. Note that [U0] = [W₀⁺(λ)] =1 in the Grothendieck ring.

Let us denote by U₋^±_s₋_2m:=C· |±

the 1-dimensional representation with odd generator |− or even generator |+.

(HereU_p⁺:=Up) We have

U_m¹⊗Un²U_m¹₊²_n. (32)

Let us introduce the parity elementσ:= [U₀⁻] in the Grothendieck ring. Then U₀⁻⊗W_s^±W_s^∓, U₀⁻⊗U_p^±U^∓_p. (33) Hence

σ[Ws^±] = [Ws^∓], σ[U^±_p] = [U^∓_p], σ²=1, (34) and we can rewrite in the Grothendieck ring:

[ρ+(q

s 2+¹₂

∗ λ)][ρ−(q⁻

s 2−¹₂

∗ λ)] =

∞ m=0

[U₋s−2m⊗Ws⁽⁻¹⁾^m(λ)]

=

∞ m=0

σ^m[U₋s−2m][Ws⁺(λ)]

= [Ws⁺(λ)]

∞ m=0

σ^m[U₋s−2m]

= [Ws⁺(λ)] ·fs

where fs:=

∞ m=0

σ^m[U₋s−2m] = [U₋s]

∞ m=0

σ^m[U₋₂]^m= [U₋s]

1−σ[U₋₂]. (35) For simplicity, let us always fix the highest weight of the finite-dimensional module to be even and rewrite Ws(λ):=W_s⁺(λ).

Now from previous observation,

[Ws⁺(λ)] = [Ws(λ)] + [W₋⁽⁻¹⁾_s₋₂^s⁺¹(λ)] = [Ws(λ)] +σ^s⁺¹[W₋⁺_s₋₂(λ)].