Rigidity of some classes of Lie algebras in connection to Leibniz algebras

TL;DR

This paper investigates the rigidity properties of Lie and Leibniz algebras, exploring how rigidity concepts relate between these algebraic structures and identifying conditions for Leibniz rigidity.

Contribution

It analyzes the connections between various rigidity notions in Lie and Leibniz algebras and provides necessary conditions for Lie algebras to be Leibniz rigid.

Findings

Identifies relations between different types of rigidity in Lie and Leibniz algebras

Provides necessary conditions for Lie algebras to exhibit Leibniz rigidity

Enhances understanding of algebraic varieties of Lie and Leibniz algebras

Abstract

In this paper we focus on algebraic aspects of contractions of Lie and Leibniz algebras. The rigidity of algebras plays an important role in the study of their varieties. The rigid algebras generate the irreducible components of this variety. We deal with Leibniz algebras which are generalizations of Lie algebras. In Lie algebras case, there are different kind of rigidities (rigidity, absolutely rigidity, geometric rigidity and e.c.t.). We explore the relations of these rigidities with Leibniz algebra rigidity. Necessary conditions for a Lie algebra to be Leibniz rigid are discussed.