The full cohomology, abelian extensions and formal deformations of Hom-pre-Lie algebras

TL;DR

This paper develops a comprehensive cohomology theory for Hom-pre-Lie algebras, linking it to abelian extensions and deformations, thereby advancing the understanding of their algebraic structure and classification.

Contribution

It introduces a full cohomology framework for Hom-pre-Lie algebras that incorporates the Hom-type structure and connects to deformations and extensions.

Findings

Second cohomology classifies abelian extensions.

Cohomology aligns with simultaneous deformations of multiplication and homomorphism.

Provides a new cohomology theory tailored for Hom-pre-Lie algebras.

Abstract

The main purpose of this paper is to provide a full cohomology of a Hom-pre-Lie algebra with coefficients in a given representation. This new type of cohomology exploit strongly the Hom-type structure and fits perfectly with simultaneous deformations of the multiplication and the homomorphism defining a Hom-pre-Lie algebra. Moreover, we show that its second cohomology group classifies abelian extensions of a Hom-pre-Lie algebra by a representation.