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SEMISIMPLICITY OF HECKE AND (WALLED) BRAUER ALGEBRAS

Part of: Representation theory of groups Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  25 October 2016

HENNING HAAHR ANDERSEN ,
CATHARINA STROPPEL  and
DANIEL TUBBENHAUER
HENNING HAAHR ANDERSEN
Affiliation:
Aarhus University, Centre for Quantum Geometry of Moduli Spaces, Ny Munkegade 118, Building 1530, Room 327, 8000 Aarhus C, Denmark email mathha@qgm.au.dk
CATHARINA STROPPEL
Affiliation:
Universität Bonn, Mathematisches Institut, Endenicher Allee 60, Room 4.007, 53115 Bonn, Germany email stroppel@math.uni-bonn.de
DANIEL TUBBENHAUER*
Affiliation:
Universität Bonn, Mathematisches Institut, Endenicher Allee 60, Room 1.003, 53115 Bonn, Germany email dtubben@math.uni-bonn.de
*
✉dtubben@math.uni-bonn.de
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Abstract

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We show how to use Jantzen’s sum formula for Weyl modules to prove semisimplicity criteria for endomorphism algebras of $\mathbf{U}_{q}$-tilting modules (for any field $\mathbb{K}$ and any parameter $q\in \mathbb{K}-\{0,-1\}$). As an application, we recover the semisimplicity criteria for the Hecke algebras of types $\mathbf{A}$ and $\mathbf{B}$, the walled Brauer algebras and the Brauer algebras from our more general approach.

Keywords

quantum groups(quantum) tilting modulesJantzen’s sum formulassemisimplicityHecke and Brauer algebras

MSC classification

Primary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations
Secondary: 17B10: Representations, algebraic theory (weights) 20C08: Hecke algebras and their representations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 103 , Issue 1 , August 2017 , pp. 1 - 44
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

Footnotes

Elements of the text in the article appear in colour online available at 10.1017/S1446788716000392.

H.H.A. was supported by the Center of Excellence grant ‘Centre for Quantum Geometry of Moduli Spaces (QGM)’ from the Danish National Research Foundation (DNRF), C.S. by a Hirzebruch professorship of the Max-Planck-Gesellschaft and D.T. by a research funding of the Deutsche Forschungsgemeinschaft (DFG) during this work.

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