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CATEGORIES OF MODULES OVER AN AFFINE KACMOODY ALGEBRA AND FINITENESS OF THE KAZHDANLUSZTIG
 

Summary: CATEGORIES OF MODULES OVER AN AFFINE KAC­MOODY
ALGEBRA AND FINITENESS OF THE KAZHDAN­LUSZTIG
TENSOR PRODUCT
MILEN YAKIMOV
Abstract. To each category C of modules of finite length over a complex
simple Lie algebra g, closed under tensoring with finite dimensional modules,
we associate and study a category AFF(C) of smooth modules (in the sense
of Kazhdan and Lusztig [13]) of finite length over the corresponding affine
Kac­Moody algebra in the case of central charge less than the critical level.
Equivalent characterizations of these categories are obtained in the spirit of the
works of Kazhdan­Lusztig [13] and Lian­Zuckerman [18, 19]. In the main part
of this paper we establish a finiteness result for the Kazhdan­Lusztig tensor
product which can be considered as an affine version of a theorem of Kostant
[17]. It contains as special cases the finiteness results of Kazhdan, Lusztig
[13] and Finkelberg [7], and states that for any subalgebra f of g which is
reductive in g the "affinization" of the category of finite length admissible (g, f)
modules is stable under Kazhdan­Lusztig's tensoring with the "affinization"
of the category of finite dimensional g modules (which is O in the notation
of [13, 14, 15]).
1. Introduction

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics