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Classification of five-dimensional symmetric Leibniz algebras

Iroda Choriyeva, Abror Khudoyberdiyev

TL;DR

The paper addresses the complete classification of five-dimensional complex solvable symmetric Leibniz algebras. It uses the structural description of symmetric Leibniz algebras as a Lie algebra $(\mathcal{G},[-,-])$ equipped with a symmetric bilinear form $\omega: \mathcal{G}\times\mathcal{G}\to Z(\mathcal{G})$ satisfying orthogonality conditions, assembling algebras from the cases determined by the center and the splitting of the underlying Lie algebra. The main result shows infinitely many isomorphism classes, enumerated as $30$ one-parameter families, $8$ two-parameter families, $1$ three-parameter family, and $38$ additional isomorphism classes, with no exists non-solvable, non-split, non-Lie $5$-dimensional symmetric Leibniz algebras. This work thus provides a complete atlas for $5$-dimensional complex symmetric Leibniz algebras, extending low-dimensional classifications and enabling systematic study of their automorphisms and invariant structures.

Abstract

In this paper we give the complete classification of $5$-dimensional complex solvable symmetric Leibniz algebras.

Classification of five-dimensional symmetric Leibniz algebras

TL;DR

The paper addresses the complete classification of five-dimensional complex solvable symmetric Leibniz algebras. It uses the structural description of symmetric Leibniz algebras as a Lie algebra equipped with a symmetric bilinear form satisfying orthogonality conditions, assembling algebras from the cases determined by the center and the splitting of the underlying Lie algebra. The main result shows infinitely many isomorphism classes, enumerated as one-parameter families, two-parameter families, three-parameter family, and additional isomorphism classes, with no exists non-solvable, non-split, non-Lie -dimensional symmetric Leibniz algebras. This work thus provides a complete atlas for -dimensional complex symmetric Leibniz algebras, extending low-dimensional classifications and enabling systematic study of their automorphisms and invariant structures.

Abstract

In this paper we give the complete classification of -dimensional complex solvable symmetric Leibniz algebras.
Paper Structure (6 sections, 8 theorems, 91 equations)

This paper contains 6 sections, 8 theorems, 91 equations.

Table of Contents

  1. Preliminaries
  2. Classification of symmetric Leibniz algebras up to dimension four
  3. Classification of five-dimensional solvable symmetric Leibniz algebras
  4. Five-dimensional solvable symmetric Leibniz algebras, whose underlying Lie algebra is non-split
  5. Five-dimensional solvable symmetric Leibniz algebras, whose underlying Lie algebra is $\mathcal{S}_{4} \oplus \mathbb{C}$
  6. Five-dimensional solvable symmetric Leibniz algebras, whose underlying Lie algebra is split with more than one-dimensional direct factor

Key Result

Proposition 1.3

Barreiro Let $(\mathcal{L}, \cdot)$ be an algebra. The following assertions are equivalent:

Theorems & Definitions (18)

  • Remark
  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Definition 1.5
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • ...and 8 more