Operad with a prescribed element

Abstract

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.