Deformations and abelian extensions on anti-pre-Lie algebras

TL;DR

This paper explores the structure and deformations of anti-pre-Lie algebras, introducing cohomology, Nijenhuis operators, and classifying abelian extensions, advancing the understanding of their algebraic properties.

Contribution

It introduces the representation theory, cohomology, and classification of abelian extensions for anti-pre-Lie algebras, providing new tools for their analysis.

Findings

Defined the second cohomology group for anti-pre-Lie algebras.

Established conditions for rigidity based on cohomology.

Classified abelian extensions via second cohomology group.

Abstract

In this paper, we introduce the representation of anti-pre-Lie algebras and give the second cohomology group of anti-pre-Lie algebras. As applications, first, we study linear deformations of anti-pre-Lie algebras. The notion of a Nijenhuis operator on an anti-pre-Lie algebra is introduced which can generate a trivial linear deformation of an anti-pre-Lie algebra. Then, we study formal deformations of anti-pre-Lie algebras. We show that the infinitesimal of a formal deformation is a 2-cocycle with the coefficients in the regular representation and depends only on its cohomology class. Moreover, if the second cohomology group $H^2(A;A)$ is trivial, then the anti-pre-Lie algebra is rigid. Finally, we introduce the notion of abelian extensions. We show that abelian extensions are classified by the second cohomology group $H^2(A;V)$.