α-TYPE CHEVALLEY–EILENBERG COHOMOLOGY OF HOM-LIE ALGEBRAS AND BIALGEBRAS

Abstract

The purpose of this paper is to define an α-type cohomology, which we call α-type Chevalley-Eilenberg cohomology, for Hom-Lie algebras. We relate it to the known Chevalley-Eilenberg cohomology and provide explicit computations for some examples. Moreover, using this cohomology we study formal deformations of Hom-Lie algebras, where the bracket as well as the structure map α are deformed. Furthermore, we provide a generalization of the Grand Crochet and study, in a particular case, the α-type cohomology for Hom-Lie bialgebras.

Notes on formal deformations of quantum planes and universal enveloping algebras

“…One-parameter formal hom-associative deformations and one-parameter formal hom-Lie deformations were first introduced by Makhlouf and Silvestrov in [7], and later expanded on by Ammar, Ejbehi and Makhlouf in [10], and then by Hurle and Makhlouf in [11]. The idea behind these kinds of deformations is to deform not only the multiplication map or the Lie bracket, but also the twisting map α, resulting also in a deformation of the twisted associativity condition and the twisted Jacobi identity, respectively.…”

Cohomology and deformations of O-operators on Hom-associative algebras

“…A huge research activity was dedicated to Hom-type algebras, due in part to the prospect of having a general framework in which one can produce many types of natural deformations which are of interest to both mathematicians and physicists. Cohomology and deformations of Hom-Lie and Hom-associative algebras were studied in [4,29,43]. Notice that Cohomology and deformations of O-operators on Hom-Lie algebras were discussed in [3].…”

Equivariant formal deformations of Hom-pre-Lie algebras

“…Later, following his ideas, deformation theory for various types of algebras has been studied, see [2,11,[17][18][19][20]. Hurle and Makhlouf studied cohomology and deformation theory for Hom-associative and Hom-Lie algebras in [8,9]. In [16], Mukherjee and Saha studied cohomology and deformation theory for Hom-Leibniz algebras.…”