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Regular evolution algebras are closed under subalgebras

Manuel Ladra, Andrés Pérez-Rodríguez

Abstract

The main goal of this note is to show that subalgebras of regular evolution algebras are themselves evolution algebras. This allows us to assume, without loss of generality, that every subalgebra in the regular setting has a basis consisting of vectors with disjoint supports. Finally, we use this result to characterise the existence of codimension-one subalgebras in regular evolution algebras.

Regular evolution algebras are closed under subalgebras

Abstract

The main goal of this note is to show that subalgebras of regular evolution algebras are themselves evolution algebras. This allows us to assume, without loss of generality, that every subalgebra in the regular setting has a basis consisting of vectors with disjoint supports. Finally, we use this result to characterise the existence of codimension-one subalgebras in regular evolution algebras.

Paper Structure

This paper contains 1 section, 5 theorems, 16 equations.

Table of Contents

  1. Acknowledgements

Key Result

Lemma 1

Let $\mathcal{E}$ be a regular evolution algebra with natural basis $B=\{e_1,\dots,e_n\}$ and structure matrix $M_B(\mathcal{E})$. Then, the one-dimensional subspace $\mathop{\mathrm{span}}\nolimits\{\alpha_1e_1+\dots+\alpha_ne_n\}$ is a proper nonzero subalgebra of $\mathcal{E}$ if and only if the

Theorems & Definitions (14)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Remark 3
  • Corollary 4
  • proof
  • Proposition 5
  • proof
  • Remark 6
  • ...and 4 more