A criterion for the Jacobson semisimplicity of the green ring of a finite tensor category

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.