2002/10/11 by Milen Yakimov, Yakimov, Milen
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math-ph #math.MP #math.QA #math.RT
paper · pdf · doi:10.48550/arxiv.math/0210180
19 pages, AMS-Latex, v2 contains several minor changes
openalex publication_date 2002/10/11 · arxiv created 2004/10/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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)κof smooth modules (in the sense of Kazhdan and Lusztig [KL1]) of finite length over the corresponding affine Kac-Moody 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 Kazhdan-Lusztig [KL1] and Lian-Zuckerman [LZ1]. In the main part of this paper we establish a finiteness result for the Kazhdan-Lusztig tensor product which can be considered as an affine version of a theorem of Kostant [K]. It contains as special cases the finiteness results of Kazhdan, Lusztig [KL] and Finkelberg [F], 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 Kazhdan-Lusztig's tensoring with the "affinization" of the category of finite dimensional g modules (which is Oκin the notation of [KL1, KL2, KL3]).