Cohomologies and Deformations of Generalized Left-symmetric Algebras

TL;DR

This paper develops cohomology and deformation theories for generalized left-symmetric algebras, extending existing theorems and classifying simple superalgebras through infinitesimal deformations.

Contribution

It introduces generalized cohomology and deformation frameworks and generalizes a key theorem linking right-symmetric and Chevalley-Eilenberg cohomologies.

Findings

Established generalized cohomology and deformation theories.

Generalized a theorem relating right-symmetric and Chevalley-Eilenberg cohomologies.

Classified all 3-dimensional complex simple left-symmetric superalgebras.

Abstract

The purpose of this paper is to develop a cohomology and deformation theories for generalized left-symmetric algebras.We introduce the notions of generalized left-symmetric cohomology and deformation. We also generalize a theorem of Dzhumadil'daev on connections between the right-symmetric cohomology and Chevalley-Eilenberg cohomology. As an application, we obtain a factorization theorem in left-symmetric superalgebras cohomology. Finally, we obtain all complex simple left-symmetric superalgebras of dimension 3 by the infinitesimal deformations of a given left-symmetric superalgebras.