On classification of m-dimensional algebras

U Bekbaev

doi:10.1088/1742-6596/819/1/012012

Abstract

A constructive approach to the classification and invariance problems, with respect to basis changes, of the finite dimensional algebras is offered. A construction of an invariant open, dense (in the Zariski topology) subset of the space of structure constants of algebras is given. A classification of all algebras with structure constants from this dense set is given by providing canonical representatives of their orbits. A finite system of generators for the corresponding field of invariant rational functions of structure constants is shown.

Highlights

The classification of finite dimensional algebras is an important problem in Algebra
In general for the given dimension we provide a method how to construct an invariant, open, dense subset of the space of structure constants of algebras and classify all algebras who’s systems of structure constants are from this set
We provide a basis for the field of invariant rational functions of structure constants as well

Summary

Introduction

The classification of finite dimensional algebras is an important problem in Algebra. The classification of finite dimensional simple and semi-simple associative algebras by Wedderburn, the classification of finite dimensional simple and semi-simple Lie algebras by Cartan are well known Their classifications are examples of structural (basis free, invariant) approaches to the classification problem of algebras. In reality one may be interested in classification of algebras with respect to specific changes of basis Another approach to the classification and invariants problems of finite dimensional algebras is coordinate (basis based, structure constants) approach. In any finite dimensional algebra case a basis based approach is considered in [4]. In this paper we consider the classification and invariants problems of finite dimensional algebras.

Preliminaries

Commutative and anti-commutative algebra cases