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Please use this identifier to cite or link to this item: http://hdl.handle.net/1942/25368

Title: A CRITERION FOR THE JACOBSON SEMISIMPLICITY OF THE GREEN RING OF A FINITE TENSOR CATEGORY
Authors: Wang, Zhihua
Li, Libin
Zhang, Yinhuo
Issue Date: 2018
Citation: GLASGOW MATHEMATICAL JOURNAL, 60(1), p. 253-272
Abstract: This paper deals with the Green ring G(C) of a finite tensor category C with finitely many indecomposable objects over an algebraically closed field k. The first part of this paper is through the Casimir number of C to determine when the Green ring G(C), or the Green algebra G(C)⊗Z K over a field K is Jacobson semisimple (namely, has zero Jacobson radical). It turns out that G(C) ⊗Z K is Jacobson semisimple if and only if the Casimir number of C is not zero in K. For the Green ring G(C) itself, G(C) is Jacobson semisimple if and only if the Casimir number of C is not zero. The second part of this paper focuses on the case where C = Rep(kG) for a cyclic group G of order p over a field k of characteristic p. In this case, the Casimir number of C is computable and is shown to be 2p 2. This leads to a complete description of the Jacobson radical of the Green algebra G(C) ⊗Z K over any field K.
URI: http://hdl.handle.net/1942/25368
DOI: 10.1017/S0017089517000246
ISSN: 0017-0895
Category: A1
Type: Journal Contribution
Appears in Collections: Research publications

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