On $\alpha$-type (equivariant) cohomology of Hom-pre-Lie algebras

TL;DR

This paper introduces a new equivariant cohomology theory for Hom-pre-Lie algebras, which accounts for structure deformations and group actions, advancing the understanding of their algebraic and deformation properties.

Contribution

It develops a natural cohomology framework for Hom-pre-Lie algebras with group actions, including deformation theory in the equivariant setting.

Findings

Defined a new cohomology controlling deformations

Formulated equivariant cohomology with group actions

Studied formal deformation theory in the equivariant context

Abstract

In this paper, we define a new cohomology theory for multiplicative Hom-pre-Lie algebras which controls deformations of Hom-pre-Lie algebra structure. This new cohomology is a natural one by considering the structure map. We develop equivariant cohomology theory for a Hom-pre-Lie algebra equipped with a finite group action by formulating a proper notion of coefficients system for the equivariant cohomology. We also study the associated formal deformation theory for Hom-pre-Lie algebras in the equivariant context.