Cohomology of BiHom-associative algebras

Abstract

Bihom-associative algebras have been recently introduced in the study of group homcategories. In this paper, we introduce a Hochschild type cohomology for bihom-associative algebras with suitable coefficients. The underlying cochain complex (with coefficients in itself) can be given the structure of an operad with a multiplication. Hence, the cohomology inherits a Gerstenhaber structure. We show that this cohomology also control corresponding formal deformations.Finally, we introduce bihom-associative algebras up to homotopy and show that some particular classes of these homotopy algebras are related to the above Hochschild cohomology.

Deformations of Loday-type algebras and their morphisms

“…With the above notations (O α,β , • ′ i , id) forms an operad. The proof of the above proposition is simple and can be found in [7,8] when O is the endomorphism operad associated to a vector space. The operad (O α,β , • ′ i , id) is called the twisted variation of O twisted by α and β.…”