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Further results on modularity in evolution algebras

Manuel Ladra, Andrés Pérez-Rodríguez

Abstract

In this paper, we study modularity in the context of evolution algebras. Although this property has been previously considered, a complete description is still missing in several natural settings. In particular, we obtain a full characterisation of modular evolution algebras in the nilpotent case and in the class of supersolvable regular evolution algebras.

Further results on modularity in evolution algebras

Abstract

In this paper, we study modularity in the context of evolution algebras. Although this property has been previously considered, a complete description is still missing in several natural settings. In particular, we obtain a full characterisation of modular evolution algebras in the nilpotent case and in the class of supersolvable regular evolution algebras.

Paper Structure

This paper contains 4 sections, 8 theorems, 21 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Completeness as a sufficient condition for modularity
  4. Modularity in a specific case of regular evolution algebras

Key Result

Proposition 2.1

Let $\mathcal{E}$ be a finite-dimensional solvable (not necessarily evolution) algebra. Then, $\mathcal{E}$ is modular if and only if every subalgebra of $\mathcal{E}$ is a quasi-ideal. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Proper subalgebras and subalgebra lattice of the real evolution algebra with product $e_1^2=e_2^2=e_3$ and $e_3^2=0$.
  • Figure 2: Proper subalgebras and subalgebra lattice of $\mathcal{E}_\mathbb{C}^{\mathrm{reg}}(0,\frac{1}{4},\frac{1}{4})$.

Theorems & Definitions (21)

  • Proposition 2.1: An_94
  • Remark 2.2
  • Proposition 3.1
  • proof
  • Example 3.2: LPP_25
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Theorem 3.5
  • proof
  • ...and 11 more