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Ideals in Arbitrary Three-Dimensional Algebras

M. V. Velasco, U. A. Rozikov, B. A. Narkuziev

Abstract

In this paper, we study arbitrary (not necessarily associative) 3-dimensional algebras. Such an algebra A is determined by a basis and the corresponding multiplication table, which is specified by 27 structure constants. We describe all ideals of A, providing an explicit characterization of both 1-dimensional and 2-dimensional ideals. Moreover, we classify 2-dimensional ideals into 4 distinct types. We prove that A either has infinitely many ideals or at most 4. We also show that, in any case, the maximum number of 1-dimensional ideals is 3, while the maximum number of 2-dimensional ideals is 2. Finally, we present a class of algebras with a finite number of ideals that attain this theoretical maximum.

Ideals in Arbitrary Three-Dimensional Algebras

Abstract

In this paper, we study arbitrary (not necessarily associative) 3-dimensional algebras. Such an algebra A is determined by a basis and the corresponding multiplication table, which is specified by 27 structure constants. We describe all ideals of A, providing an explicit characterization of both 1-dimensional and 2-dimensional ideals. Moreover, we classify 2-dimensional ideals into 4 distinct types. We prove that A either has infinitely many ideals or at most 4. We also show that, in any case, the maximum number of 1-dimensional ideals is 3, while the maximum number of 2-dimensional ideals is 2. Finally, we present a class of algebras with a finite number of ideals that attain this theoretical maximum.
Paper Structure (13 sections, 18 theorems, 106 equations)

This paper contains 13 sections, 18 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Structure constants
  4. Determining the ideals of a 3-dimensional algebra
  5. One-dimensional ideals of three-dimensional algebras
  6. Two-dimensional ideals of three-dimensional algebras
  7. Characterization of 2-dimensional ideals
  8. Two dimensional ideals of $B$-type I
  9. Two dimensional ideals of $B$-type II and III
  10. Two dimensional ideals of $B$-type IV
  11. Number of ideals when type I exists
  12. A case of algebras with 2 ideals of dimension 1
  13. A class of three-dimensional algebras having 2 ideals of type IV

Key Result

Theorem 1

Let $\mathcal{A}$ be an algebra over $\mathbb{K}$, and let $B=\{e_{1},e_{2},e_{3}\}$ be a basis of $\mathcal{A}$. Let $M:=(M_{1}|M_{2}|M_{3})$ be the $3D$-matrix determining the multiplication of $\mathcal{A}$ with respect to $B.$ Let $u=\sum\limits_{i=1}^{3}u_{i}e_{i}\in \mathcal{A}$. Then $I=\math is an eigenvector of the matrices $\widehat{M}_{k}$ and $\widetilde{M}_{k}$ given in (asocia), for

Theorems & Definitions (38)

  • Definition 1
  • Theorem 1
  • proof
  • Example 1
  • Definition 2
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • Proposition 1
  • ...and 28 more