Computation of isotopisms of algebras over finite fields by means of graph invariants

TL;DR

This paper introduces graph-based functors that efficiently determine isomorphisms and isotopisms of finite-dimensional algebras over finite fields, simplifying classification tasks.

Contribution

It presents a novel graph-theoretic approach using faithful functors to identify algebra isomorphisms and isotopisms, reducing computational complexity.

Findings

Explicit classification of 2- and 3-dimensional partial quasigroup rings

Demonstration of the efficiency of graph invariants in algebra isomorphism problems

Reduction in computational cost for algebra classification

Abstract

In this paper we define a pair of faithful functors that map isomorphic and isotopic finite-dimensional algebras over finite fields to isomorphic graphs. These functors reduce the cost of computation that is usually required to determine whether two algebras are isomorphic. In order to illustrate their efficiency, we determine explicitly the classification of two- and three-dimensional partial quasigroup rings.