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On split regular Hom-Lie superalgebras

Helena Albuquerque, Elisabete Barreiro, Antonio J. Calderón, José M. Sánchez

Abstract

We introduce the class of split regular Hom-Lie superalgebras as the natural extension of the one of split Hom-Lie algebras and Lie superalgebras, and study its structure by showing that an arbitrary split regular Hom-Lie superalgebra ${\frak L}$ is of the form ${\frak L} = U + \sum_j I_j$ with $U$ a linear subspace of a maximal abelian graded subalgebra $H$ and any $I_j$ a well described (split) ideal of ${\frak L}$ satisfying $[I_j,I_k] = 0$ if $j \neq k$. Under certain conditions, the simplicity of ${\frak L}$ is characterized and it is shown that ${\frak L}$ is the direct sum of the family of its simple ideals.

On split regular Hom-Lie superalgebras

Abstract

We introduce the class of split regular Hom-Lie superalgebras as the natural extension of the one of split Hom-Lie algebras and Lie superalgebras, and study its structure by showing that an arbitrary split regular Hom-Lie superalgebra is of the form with a linear subspace of a maximal abelian graded subalgebra and any a well described (split) ideal of satisfying if . Under certain conditions, the simplicity of is characterized and it is shown that is the direct sum of the family of its simple ideals.
Paper Structure (4 sections, 15 theorems, 95 equations)

This paper contains 4 sections, 15 theorems, 95 equations.

Table of Contents

  1. Introduction and first definitions
  2. Connections of roots techniques
  3. Decompositions
  4. The simple components

Key Result

Lemma 1.1

For any $\alpha, \beta \in \Lambda \cup \{0\}$ the following assertions hold.

Theorems & Definitions (37)

  • Definition 1.1
  • Definition 1.2
  • Example 1.1
  • Lemma 1.1
  • proof
  • Remark 1.1
  • Lemma 1.2
  • proof
  • Remark 1.2
  • Definition 2.1
  • ...and 27 more