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Classification in a rotational flow of two-dimensional algebras

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

Abstract

In this paper, we examine a time-dependent family of two-dimensional algebras. We investigate the conditions under which any two algebras from this family, formed at different times, are isomorphic. Our findings reveal that the flow comprises of uncountable pairwise non-isomorphic algebras, including one commutative algebra. Additionally, we compare our results with a previously established classification of 2-dimensional real algebras.

Classification in a rotational flow of two-dimensional algebras

Abstract

In this paper, we examine a time-dependent family of two-dimensional algebras. We investigate the conditions under which any two algebras from this family, formed at different times, are isomorphic. Our findings reveal that the flow comprises of uncountable pairwise non-isomorphic algebras, including one commutative algebra. Additionally, we compare our results with a previously established classification of 2-dimensional real algebras.
Paper Structure (6 sections, 5 theorems, 79 equations, 1 figure)

This paper contains 6 sections, 5 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Cubic matrices
  3. Two-dimensional algebras over $\mathbb{R}$
  4. Flow of algebras
  5. Two-dimensional flow of algebras
  6. Comparision with Theorem 1

Key Result

Theorem 1

Bekbaev Any non-trivial 2-dimensional real algebra is isomorphic to only one of the following listed, by their matrices of structural constants, algebras: $\mathcal{A}_{13}=\left ( \right), \mathcal{A}_{14}=\left ( \right), \mathcal{A}_{15}=\left ( \right).$

Figures (1)

  • Figure 1: The partition of the time set $\{(0,t): 0\leq t\}$ corresponding to the classification of algebras in the FA $A^{[t]}$ with the matrix (\ref{['2.5']}).

Theorems & Definitions (19)

  • Theorem 1
  • Example 1
  • Remark 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 1
  • ...and 9 more