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A classification of 2-dimensional endo-commutative straight algebras of type II_1

Sin-Ei Takahasi, Kiyoshi Shirayanagi, Makoto Tsukada

Abstract

In this paper, we present a complete classification of 2-dimensional endo-commutative straight algebras of type II$_1$ over any field. An endo-commutative algebra is a non-associative algebra in which the square mapping preserves multiplication. A 2-dimensional straight algebra satisfies the condition that there exists an element $x$ such that $x$ and $x^2$ are linearly independent. The term type II$_1$ denotes a distinguishing characteristic of its structure matrix, which has rank 2. We provide multiplication tables for these algebras, listing them up to isomorphism.

A classification of 2-dimensional endo-commutative straight algebras of type II_1

Abstract

In this paper, we present a complete classification of 2-dimensional endo-commutative straight algebras of type II over any field. An endo-commutative algebra is a non-associative algebra in which the square mapping preserves multiplication. A 2-dimensional straight algebra satisfies the condition that there exists an element such that and are linearly independent. The term type II denotes a distinguishing characteristic of its structure matrix, which has rank 2. We provide multiplication tables for these algebras, listing them up to isomorphism.
Paper Structure (12 sections, 33 theorems, 150 equations)

This paper contains 12 sections, 33 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Isomorphism criterion for 2-dimensional algebras
  3. Characterization of 2-dimensional endo-commutative straight algebras
  4. 2-dimensional endo-commutative straight algebras of rank 2
  5. Characterization of type II$_1$ algebras
  6. Five equivalence relations on $K^*$
  7. Main results
  8. Classification of $(\mathcal{ECS}_{011})_{1}$
  9. Classification of $(\mathcal{ECS}_{011})_{3}$
  10. Classification of $(\mathcal{ECS}_{011})_{4}$
  11. Proof of Theorem \ref{['thm:char-not=2']}
  12. Proof of Theorem \ref{['thm:char=2']}

Key Result

Lemma 3.1

The straight algebra $S(p, q, a, b, c, d)$ is endo-commutative iff the point $(p, q, a, b, c, d)\in K^6$ satisfies

Theorems & Definitions (49)

  • Lemma 3.1
  • Lemma 3.2
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • Lemma 5.3
  • proof
  • Lemma 5.4
  • proof
  • ...and 39 more