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On the Classification of Two-Dimensional Algebras

Bekbaev U

Abstract

We provide a clarification of the classification of two-dimensional algebras over an arbitrary base field. Using this clarification, we determine the number of non-isomorphic two-dimensional algebras over a finite field.

On the Classification of Two-Dimensional Algebras

Abstract

We provide a clarification of the classification of two-dimensional algebras over an arbitrary base field. Using this clarification, we determine the number of non-isomorphic two-dimensional algebras over a finite field.
Paper Structure (3 sections, 3 theorems, 31 equations)

This paper contains 3 sections, 3 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. On the Classification of Two-Dimensional Algebras over an Arbitrary Base Field
  3. On the Number of Non-Isomorphic Two-Dimensional Algebras over a Finite Field

Key Result

Theorem 1.1

Any non-trivial two-dimensional algebra over a field $\mathbb{F}$ with $char(\mathbb{F})\neq 2,3$ is isomorphic to exactly one of the following algebras, given by their matrices of structure constants. Any non-trivial 2-dimensional algebra over a field $\mathbb{F}$, $char.(\mathbb{F})= 2$, is isomorphic to only one of the following listed, by their matrices of structure constants, algebras. Moreo

Theorems & Definitions (6)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • Remark 3.3