Lie algebras with associative structures. Applications to the study of 2-step nilpotent Lie algebras

TL;DR

This paper explores Lie algebras with associative structures, focusing on 2-step nilpotent Lie algebras, their deformations, and cohomology to understand their algebraic properties and classifications.

Contribution

It introduces a framework for Lie algebras with associative brackets and analyzes their deformations and cohomology, specifically for 2-step nilpotent cases.

Findings

Characterization of nilpotent Lie algebras with associative brackets.

Computation of cohomology groups for deformation analysis.

Insights into the structure and classification of 2-step nilpotent Lie algebras.

Abstract

We investigate Lie algebras whose Lie bracket is also an associative or cubic associative multiplication to characterize the class of nilpotent Lie algebras with a nilindex equal to 2 or 3. In particular we study the class of 2-step nilpotent Lie algebras, their deformations and we compute the cohomology which parametrize the deformations in this class.