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

Journal Contribution - Journal Article

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.
Journal: GLASGOW MATHEMATICAL JOURNAL
ISSN: 0017-0895
Issue: 1
Volume: 60
Pages: 253 - 272
Publication year:2018
Keywords:finite tensor category, green ring, Casimir number, Jacobson radical, Frobenius algebra.
BOF-keylabel:yes
IOF-keylabel:yes
BOF-publication weight:0.1
CSS-citation score:1
Authors:International
Authors from:Higher Education
Accessibility:Open