Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-degenerate evolution algebras

Antonio Jesús Calderón Martín, Amir Fernández Ouaridi, Ivan Kaygorodov

Abstract

In this paper we introduce a new invariant for a non-degenerate evolution algebra, which consists of an ordered sequence of evolution algebras of lower dimension, belonging all of them to a specific family. We use this invariant to propose a method to classify non-degenerate evolution algebras, and we apply it up to dimension 3. We also use it to describe the derivations of some families of evolution algebras and the variety of evolution algebras with square not greater than 1.

Non-degenerate evolution algebras

Abstract

In this paper we introduce a new invariant for a non-degenerate evolution algebra, which consists of an ordered sequence of evolution algebras of lower dimension, belonging all of them to a specific family. We use this invariant to propose a method to classify non-degenerate evolution algebras, and we apply it up to dimension 3. We also use it to describe the derivations of some families of evolution algebras and the variety of evolution algebras with square not greater than 1.
Paper Structure (26 sections, 22 theorems, 80 equations)

This paper contains 26 sections, 22 theorems, 80 equations.

Table of Contents

  1. Definitions and preliminary results
  2. The $\delta$-index of a non-degenerate evolution algebra
  3. The $\Delta$-trace of a non-degenerate evolution algebra
  4. The classification method
  5. Classification of 2-dimensional non-degenerate evolution algebras
  6. Case Lg
  7. Subcase $\Delta({\mathbb A})=({\bf O}_1, {\bf O}_1)$
  8. Subcase $\Delta({\mathbb A})=({\bf O}_1, {\bf E}_1)$
  9. Subcase $\Delta({\mathbb A})=({\bf E}_1, {\bf E}_1)$
  10. Case Lg
  11. Classification of 3-dimensional non-degenerate evolution algebras
  12. Case Lg
  13. Subcase $\Delta({\mathbb A})=({\bf O}_1, {\bf O}_1, {\bf O}_1)$
  14. Subcase $\Delta({\mathbb A})=({\bf O}_1, {\bf O}_1, {\bf E}_1)$
  15. Subcase $\Delta({\mathbb A})=({\bf O}_1, {\bf E}_1, {\bf E}_1)$
  16. ...and 11 more sections

Key Result

Theorem 5

Let ${\mathbb A}$ be an evolution algebra and let ${\bf B}=\left\{e_1,\ldots, e_n\right\}$ be a natural basis of ${\mathbb A}$ with structure matrix ${\bf M}_{\bf B}({\mathbb A})$. Then for another natural basis ${\bf B}'=\left\{f_{1},\ldots, f_{n}\right\}$ of ${\mathbb A}$ and ${\bf C} =(c_{ij})$ c where ${\bf C}^{(2)}=(c_{ij}^2)$.

Theorems & Definitions (54)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 4
  • Theorem 5: CSV2t
  • Theorem 6: NYM, Theorem 2.11
  • Definition 7
  • Lemma 8
  • proof
  • Definition 9
  • ...and 44 more