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

WANG, ZHIHUA ; LI, LIBIN ; ZHANG, YINHUO

Glasgow mathematical journal, Tome 60 (2018) no. 1, pp. 253-272

Résumé

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.

Détail

DOI : 10.1017/S0017089517000246