Semisimple Group (and Loop) Algebras over Finite Fields

TL;DR

This paper analyzes the structure of semisimple group and loop algebras over finite fields, providing classifications, decompositions, and counterexamples to isomorphism problems.

Contribution

It determines the structure of semisimple group algebras over specific fields and applies these results to loop algebras and isomorphism questions.

Findings

Identified minimal simple component decompositions over certain finite fields

Classified loop algebras of indecomposable RA loops

Provided counterexamples to isomorphism over various fields

Abstract

We determine the structure of the semisimple group algebra of certain groups over the rationals and over those finite fields where the Wedderburn decompositions have the least number of simple components. We apply our work to obtain similar information about the loop algebras of indecomposable RA loops and to produce negative answers to the isomorphism problem over various fields.