Narrow Results

We study operad with a prescribed element (of its underlying degree one component), generalize some of the classical results of Gerstenhaber and Voronov [M. Gerstenhaber and A. A. Voronov, Homotopy G-algebras and moduli space operad, Int. Math. Res. Not.3 (1995) 141–153]. In particular, we introduce and show that Hom–Loday algebra cohomologies carry such structures when the Hom-structure twist is an idempotent.

Bihom-associative algebras have been recently introduced in the study of group hom-categories. 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.