The Drinfeld realization of the elliptic quantum group Bq,λ(A2(2))

Kojima, Takeo; Konno, Hitoshi

doi:10.1063/1.1767296

Cited by 9 publications

(15 citation statements)

References 23 publications

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“…For the cases g being the twisted affine Lie algebras A (2) 2n and A (2) 2n−1 , Kuniba derived elliptic solutions to the face type YBE. His construction is based on a common structure of the R matrices of the twisted U q (g) to those of the B 2 case was investigated in [7]. In view of these facts, we expect the same statement as Theorem 5.3 is valid in the twisted cases, too.…”

Section: J=0mentioning

confidence: 54%

“…We have done this for the vector representations of A q,p ( sl 2 ) and B q,λ ( sl 2 ), which led to Baxter's elliptic R matrix and AndrewsBaxter-Forrester's elliptic face weights, respectively [23]. The same checks for the face weights were also done in the cases g = A (1) n , A (2) 2 [6,7]. The aim of this paper is to overcome this disadvantage by clarifying the following point concerning the face type.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we consider a problem analogous to 1). As for developments in the direction 2), we refer the reader to the papers [4,5,6,7,8]. We are especially interested in the two dimensional exactly solvable lattice models.…”

Section: Introductionmentioning

confidence: 99%

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Dynamical R Matrices of Elliptic Quantum Groups and Connection Matrices for the q-KZ Equations

Konno¹

2006

SIGMA

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Abstract. For any affine Lie algebra g, we show that any finite dimensional representation of the universal dynamical R matrix R(λ) of the elliptic quantum group B q,λ (g) coincides with a corresponding connection matrix for the solutions of the q-KZ equation associated with U q (g). This provides a general connection between B q,λ (g) and the elliptic face (IRF or SOS) models. In particular, we construct vector representations of R(λ) for g = A(1)n , and show that they coincide with the face weights derived by Jimbo, Miwa and Okado. We hence confirm the conjecture by Frenkel and Reshetikhin.

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Section: J=0mentioning

confidence: 54%

Section: Introductionmentioning

confidence: 99%

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Dynamical R Matrices of Elliptic Quantum Groups and Connection Matrices for the q-KZ Equations

Konno¹

2006

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“…Moreover, quantum W -algebra W q,p (sl(M |N )) will arise as fusion of the vertex operators for the elliptic algebra. In the case g = sl N , A (2) in this paper [9,10,14,15,16,17]. The quantum W -algebras associated with g = sl N , A…”

Section: Discussionmentioning

confidence: 95%

Commutation relations of vertex operators for Uq(sl^(M|N))

Kojima

2018

Journal of Mathematical Physics

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We consider commutation relations and invertibility relations of vertex operators for the quantum affine superalgebra Uq( sl(M |N )) by using bosonization. We show that vertex operators give a representation of the graded Zamolodchikov-Faddeev algebra by direct computation. Invertibility relations of type-II vertex operators for N > M are very similar to those of type-I for M > N . IntroductionVertex operators and corner transfer matrices are a useful tool in solvable lattice models [1,2,3]. They can be very effective way of calculating correlation functions. In the thermodynamic limit, a half transfer matrix becomes a type-I vertex operator Φ µV λ (z) of the quantum affine algebra U q (g). A type-I vertex operator is, by definition, an intertwiner of theand V (µ) are highest weight representations and V z denotes the evaluation representation [4]. In this paper we consider commutation relations and invertibility relations of the vertex operators for U q ( sl(M |N )) (M = N, M, N ≥ 1) by using bosonization [5]. We show that the vertex operators give a representation of the graded Zamolodchikov-Faddeev algebra by direct computation. Our commutation relations of the vertex operators give a higher-rank generalization of those for U q ( sl(M |1)) [6,7]. A type-. We note that the invertibility relations of the type-II vertex operators for N > M are very similar to those of the type-I for M > N . Our direct computation can be applied to bosonization of vertex operators and a L-operator for the elliptic algebra U q,p ( sl(M |N )) [8,9,10]. Moreover, quantum W -algebra W q,p (sl(M |N )) will arise as fusion of vertex operators for the elliptic algebra [11].The text is organized as follows. In Section 2 we recall bosonization of the quantum affine superalgebra U q ( sl(M |N )) and the vertex operators. In Section 3 we introduce the R-matrix and describe the main theorems. In Section 4 we give a direct proof of the main theorems. In Section 5 we discuss related topics. In Appendix A we summarize normal ordering rules of bosonic operators.where we set δ(z) = m∈Z z m . Here we use the generating functions(2.24)The Chevalley generators are obtained by h i = H i 0 (i = 1, 2, · · · , M + N − 1), (2.25) e i = X +,i 0 , f i = X −,i 0 (i = 1, 2, · · · , M + N − 1), (2.26)

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“…2 , sl(M |N ), osp(2|2) (2) [3,6,4,5,7,8,9,10]. Using the dressing method developed in non-twisted algebra [11] and twisted algebra A (2) 2 [12], we have bosonizations of the elliptic algebra U q,p (g) for g = (ADE) (1) , (BC) (1) , G (1) 2 and A (2) 2 .…”

Section: Introductionmentioning

confidence: 99%