Multiplicities and dimensions in enveloping tensor categories

TL;DR

This paper computes tensor product multiplicities and categorical dimensions in enveloping tensor categories derived from Mal'cev categories, providing explicit structural results and the Grothendieck ring.

Contribution

It introduces explicit calculations of multiplicities and dimensions in tensor categories constructed from Mal'cev categories, extending previous structural work.

Findings

Calculated tensor product multiplicities for simple objects.

Determined categorical dimensions of simple objects.

Derived the Grothendieck ring structure.

Abstract

In the previous paper arxiv:math/0610552 semisimple tensor categories were constructed out of certain regular Mal'cev categories. In this paper, we calculate the tensor product multiplicities and the categorical dimensions of the simple objects. This yields also the Grothendieck ring. The main tool is the subquotient decomposition of the basic objects.