# On Hom-Lie antialgebra

## TL;DR

This paper introduces Hom-Lie antialgebras, explores their representations and cohomology, and establishes foundational results on extensions, deformations, and Nijenhuis operators for this new algebraic structure.

## Contribution

It is the first to define Hom-Lie antialgebras and study their cohomology, extensions, deformations, and Nijenhuis operators, expanding the theory of Hom-algebra structures.

## Key findings

- Equivalent classes of abelian extensions correspond to second cohomology groups.
- 1-parameter infinitesimal deformations are characterized by 2-cocycles.
- Nijenhuis operators describe trivial deformations.

## Abstract

In this paper, we introduced the notion of Hom-Lie antialgebras. The representations and cohomology theory of Hom-Lie antialgebras are investigated. We prove that the equivalent classes of abelian extensions of Hom-Lie antialgebras are in one-to-one correspondence to elements of the second cohomology group. We also prove that 1-parameter infinitesimal deformation of a Hom-Lie antialgebra are characterized by 2-cocycles of this Hom-Lie antialgebra with adjoint representation in itself. The notion of Nijenhuis operators of Hom-Lie antialgebra is introduced to describe trivial deformations.