Deformations and Extensions of BiHom-alternative algebras

TL;DR

This paper explores the structure, cohomology, deformations, and extensions of BiHom-alternative algebras, a generalization of alternative algebras, and investigates their derivations and related properties.

Contribution

It introduces the cohomology and deformation theory for BiHom-alternative algebras and studies their extensions and derivations, expanding the understanding of their algebraic structure.

Findings

Cohomology theory for BiHom-alternative algebras developed.

Formal deformation theory for these algebras established.

Relationships between extensions and cohomology analyzed.

Abstract

The aim of this paper is to deal with BiHom-alternative algebras which are a generalization of alternative and Hom-alternative algebras, their structure is defined with two commuting multiplicative linear maps. We study cohomology and one-parameter formal deformation theory of left BiHom-alternative algebras. Moreover, we study central and $T_\theta$-extensions of BiHom-alternative algebras and their relationship with cohomology. Finally, we investigate generalized derivations and give some relevant results.