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On three-dimensional associative algebras

U. Bekbaev, I. Rakhimov

Abstract

This paper addresses the classification problem of associative algebras over arbitrary base fields. We present a list of three-dimensional associative algebras with canonical representatives of the isomorphism classes for fields of characteristic different from two and three. We also compare our lists with the most recent classifications over the complex numbers and with the nilpotent case over arbitrary base fields in dimension three, adding some comments.

On three-dimensional associative algebras

Abstract

This paper addresses the classification problem of associative algebras over arbitrary base fields. We present a list of three-dimensional associative algebras with canonical representatives of the isomorphism classes for fields of characteristic different from two and three. We also compare our lists with the most recent classifications over the complex numbers and with the nilpotent case over arbitrary base fields in dimension three, adding some comments.

Paper Structure

This paper contains 6 sections, 3 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. List of three-dimensional associative algebras over a base field $\mathbb{F}$ $(Char(\mathbb{F})\neq 2,3)$
  4. List of three-dimensional complex associative algebras
  5. List of three-dimensional permutative algebras
  6. Acknowledgments

Key Result

Theorem 1

Every non-trivial $2$-dimensional associative algebra over a field $\mathbb{F}$$(Char(\mathbb{F})\neq 2)$ is isomorphic to one of the following presented by their matrices of structure constants pairwise non-isomorphic algebras:

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Theorem 2
  • proof
  • Definition 4
  • Corollary 6.1