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

  • ZHIHUA WANG (a1) (a2), LIBIN LI (a3) and YINHUO ZHANG (a4)

Abstract

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 2p 2. 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.

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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
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