Gerstenhaber algebra structure on the cohomology of a hom-associative algebra

Abstract

A hom-associative algebra is an algebra whose associativity is twisted by an algebra homomorphism. In this paper, we define a cup product on the cohomology of a hom-associative algebra. We show that the cup product together with the degree −1 graded Lie bracket (which controls the deformation of the hom-associative algebra structure) on the cohomology forms a Gerstenhaber algebra. This generalizes a classical fact that the Hochschild cohomology of an associative algebra carries a Gerstenhaber algebra structure.2010 Mathematics Subject Classification. 16E40, 17A30.

Hom-associative algebras up to homotopy

“…Recently, the present author shows that the Hochschild cohomology H ‚ α pA, Aq of a hom-associative algebra also carries a similar structure [5,6]. More precisely, given a hom-associative algebra pA, µ, αq, there is a degree´1 graded Lie bracket on C ‚ α pA, Aq given by rf, gs " f˝g´p´1q pp´1qpq´1q g˝f, for f P C p α pA, Aq, g P C q α pA, Aq, where pf˝gq is given by pf˝gqpa 1 , .…”

“…To extend the formal deformation theory from associative algebras to hom-associative algebras, the authors in [2,19] introduce a Hochschild type cohomology theory (suitably twisted by α) for hom-associative algebras. Recently, the present author shows that like the classical associative case [8], the Hochschild cohomology of a hom-associative algebra carries a Gerstenhaber algebra structure [5]. This Gerstenhaber structure on cohomology is in fact induced from a homotopy G-algebra structure on the Hochschild cochain groups [6].…”

“…Recently, the present author shows that like the classical associative case [8], the Hochschild cohomology of a hom-associative algebra carries a Gerstenhaber algebra structure [5]. This Gerstenhaber structure on cohomology is in fact induced from a homotopy G-algebra structure on the Hochschild cochain groups [6].…”

“…Like classical case, these cohomology theory control the deformation of hom-associative algebras [2]. Recently, the present author showed that the cohomology H ‚ α pA, Aq inherits a Gerstenhaber algebra structure [5,6]. A graded hom-associative algebra consists of a graded vector space A " 'A i together with a bilinear map µ : A b A Ñ A of degree 0 and a linear map α : A Ñ A of degree 0 satisfying αpµpa, bqq " µpαpaq, αpbqq and the hom-associativity condition µpαpaq, µpb, cqq " µpµpa, bq, αpcqq, for all a, b, c P A.…”

“…In this section, we recall hom-associative algebras and hom-Lie algebras in the differential graded perspectives. For more details on hom-algebras, see [2,5,6,11,[17][18][19]24].…”