Semisimplicity of the deformations of the subcharacter algebra of an abelian group

TL;DR

This paper investigates the structure and semisimplicity of deformations of the subcharacter algebra of finite abelian groups, providing classifications and conditions for semisimplicity.

Contribution

It characterizes simple modules of certain deformations and identifies when these deformations are semisimple, especially for cyclic groups of prime order.

Findings

Deformations satisfying a non-degeneracy condition have describable simple modules.

The natural number inclusion deformation is not semisimple for finite abelian groups.

Complete classification of semisimple deformations for cyclic groups of prime order.

Abstract

For those deformations that satisfy a certain non-degeneracy condition, we describe the structure of certain simple modules of the deformations of the subcharacter algebra of a finite group. For finite abelian groups, we prove that the deformation given by the inclusion of the natural numbers, which corresponds to the algebra generated by the fibred bisets over a field of characteristic zero, is not semisimple. In the cyclic group of prime order case, we provide a complete description of the semisimple deformations.