Quantum affine algebras and their representations
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.
representation category of affine quantum groups
quantum Knizhnik-Zamolodchikov equations
scientific article; zbMATH DE number 795419
scientific article