Quantum affine algebras and their representations
type: Item
instance of: scholarly article
title: Quantum affine algebras and their representations
zbMATH Open document ID: 0854.17015
DOI: 10.1016/0022-4049(95)00051-W
published in: Journal of Pure and Applied Algebra
publication date: 1997-01-22T00:00:00Z
review text: This article is a short version of the authors' paper [Sel. Math., New Ser. 1, No. 3, 537-595 (1995; Zbl 0842.17021)]. The notion of fusion in the category \({\mathcal O}\) of representations of affine Kac-Moody algebras was introduced in the framework of conformal field theory and was developed in a number of mathematical papers. D. Kazhdan and G. Lusztig (1993-94) constructed an equivalence between the ``fusion'' category for an arbitrary negative charge and the category of representations of the corresponding quantum group. The purpose of the present paper is to construct a fusion and the quasi-associativity constraint on the corresponding representation category \({\mathcal O}\) of affine quantum groups. More precisely, the categories under consideration do not carry a monoidal structure. Nevertheless it is possible to replace the so called fusion tensor product by the ``fusion action'' of the category \({\mathcal D}\) of finite-dimensional representations on the category \({\mathcal O}^+_z\) of the highest weight representations. Quasi-associativity condition for this action is proved. Details of the proof and the treatment of the related categorical material is presented in the extended version of the paper, where, in particular, the categorical meaning of the so-called quantum Knizhnik-Zamolodchikov equations is explained.
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 81R50
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 17B67
zbMATH DE Number: 795419
zbMATH Keywords: quantum affine algebra
zbMATH Keywords: representation category of affine quantum groups
zbMATH Keywords: fusion
zbMATH Keywords: quasi-associativity constraint
zbMATH Keywords: quantum Knizhnik-Zamolodchikov equations
author: David Kazhdan
author: Yan Soibelman
reviewed by: Yuri Bespalov
MaRDI profile type: Publication
cites work: Quantum affine algebras and holonomic difference equations
cites work: Tensor Structures Arising from Affine Lie Algebras. IV
cites work: Introduction to quantum groups
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description: scientific article; zbMATH DE number 795419