Low dimensional cohomology of Hom-Lie algebras and q-deformed W(2,2) algebra

TL;DR

This paper investigates the low-dimensional cohomology of Hom-Lie algebras, particularly focusing on the q-deformed W(2,2) algebra, establishing its Hom-Lie structure and computing its cohomology groups and derivations.

Contribution

It demonstrates that the q-deformed W(2,2) algebra is a Hom-Lie algebra and computes its second cohomology group and derivations, extending classical Lie algebra results.

Findings

q-deformed W(2,2) algebra is a Hom-Lie algebra

Computed the second cohomology group of the algebra

Determined all $ ext{a}^k$-derivations and the first cohomology group

Abstract

This paper aims to study the low dimensional cohomology of Hom-Lie algebras and q-deformed W(2,2) algebra. We show that the q-deformed W(2,2) algebra is a Hom-Lie algebra. Also, we establish a one-to-one correspondence between the equivalence classes of one dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the second cohomology group of the q-deformed W(2,2) algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras. As application we compute all $\a^k$-derivations and in particular the first cohomology group of the q-deformed W(2,2) algebra.