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A CRITERION FOR THE JACOBSON SEMISIMPLICITY OF THE GREEN RING OF A FINITE TENSOR CATEGORY

Part of: Representation theory of rings and algebras Categories with structure Hopf algebras, quantum groups and related topics

Published online by Cambridge University Press:  25 September 2017

ZHIHUA WANG ,
LIBIN LI  and
YINHUO ZHANG
ZHIHUA WANG
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, China Department of Mathematics, Taizhou University, Taizhou 225300, China e-mail: mailzhihua@126.com
LIBIN LI
Affiliation:
School of Mathematical Science, Yangzhou University, Yangzhou 225002, China e-mail: lbli@yzu.edu.cn
YINHUO ZHANG
Affiliation:
Department of Mathematics and Statistics, University of Hasselt, Universitaire Campus, 3590 Diepeenbeek, Belgium e-mail: yinhuo.zhang@uhasselt.be
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Abstract

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This paper deals with the Green ring $\mathcal{G}(\mathcal{C})$ of a finite tensor category $\mathcal{C}$ with finitely many isomorphism classes of indecomposable objects over an algebraically closed field. The first part of this paper deals with the question of when the Green ring $\mathcal{G}(\mathcal{C})$, or the Green algebra $\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$K over a field K, is Jacobson semisimple (namely, has zero Jacobson radical). It turns out that $\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$K is Jacobson semisimple if and only if the Casimir number of $\mathcal{C}$ is not zero in K. For the Green ring $\mathcal{G}(\mathcal{C})$ itself, $\mathcal{G}(\mathcal{C})$ is Jacobson semisimple if and only if the Casimir number of $\mathcal{C}$ is not zero. The second part of this paper focuses on the case where $\mathcal{C}=\text{Rep}(\mathbb {k}G)$ for a cyclic group G of order p over a field $\mathbb {k}$ of characteristic p. In this case, the Casimir number of $\mathcal{C}$ is computable and is shown to be 2p2. This leads to a complete description of the Jacobson radical of the Green algebra $\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$K over any field K.

MSC classification

Primary: 16G70: Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
Secondary: 16T05: Hopf algebras and their applications 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 1 , January 2018 , pp. 253 - 272
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

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