Cohomology and Deformations of Hom-algebras

The purpose of this paper is to define cohomology structures on Hom-associative algebras and Hom-Lie algebras. The first and second coboundary maps were introduced by Makhlouf and Silvestrov in the study of one-parameter formal deformations theory. Among the relevant formulas for a generalization of Hochschild cohomology for Hom-associative algebras and a Chevalley-Eilenberg cohomology for Hom-Lie algebras, we define Gerstenhaber bracket on the space of multilinear mappings of Hom-associative algebras and Nijenhuis-Richardson bracket on the space of multilinear mappings of Hom-Lie algebras. Also we enhance the deformations theory of this Hom-algebras by studying the obstructions.

💡 Research Summary

The paper develops a comprehensive cohomology and deformation theory for Hom‑algebras, which are generalizations of associative and Lie algebras equipped with a twisting linear map α. After recalling the definitions of a Hom‑associative algebra ((A,\mu,\alpha)) and a Hom‑Lie algebra ((\mathfrak g,