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Cell 2-representations of finitary 2-categories

Part of: Categories with structure Representation theory of rings and algebras Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  29 July 2011

Volodymyr Mazorchuk  and
Vanessa Miemietz
Volodymyr Mazorchuk
Affiliation:
Department of Mathematics, Uppsala University, Box 480, SE-751 06, Uppsala, Sweden (email: mazor@math.uu.se)
Vanessa Miemietz
Affiliation:
School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK (email: v.miemietz@uea.ac.uk)
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Abstract

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We study 2-representations of finitary 2-categories with involution and adjunctions by functors on module categories over finite-dimensional algebras. In particular, we define, construct and describe in detail (right) cell 2-representations inspired by Kazhdan–Lusztig cell modules for Hecke algebras. Under some natural assumptions we show that cell 2-representations are strongly simple and do not depend on the choice of a right cell inside a two-sided cell. This reproves and extends the uniqueness result on categorification of Kazhdan–Lusztig cell modules for Hecke algebras of type A from [V. Mazorchuk and C. Stroppel, Categorification of (induced) cell modules and the rough structure of generalised Verma modules, Adv. Math. 219 (2008), 1363–1426].

Keywords

2-categorycell modulecategorificationactionfunctornatural transformationcategory with full projective functorsfinite-dimensional algebra

MSC classification

Secondary: 18D05: Double categories, $2$-categories, bicategories and generalizations 16G10: Representations of Artinian rings 17B10: Representations, algebraic theory (weights)

Information

Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 5 , September 2011 , pp. 1519 - 1545
Copyright
Copyright © Foundation Compositio Mathematica 2011

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