$L_\infty$-structures and cohomology theory of compatible $\mathcal {O}$-operators and compatible dendriform algebras

Abstract

The notion of O-operator is a generalization of Rota-Baxter operator in the presence of a bimodule over an associative algebra. A compatible O-operator is a pair consisting of two O-operators satisfying a compatibility relation. A compatible O-operator algebra is an algebra together with a bimodule and a compatible O-operator. In this paper, we construct a graded Lie algebra and an L∞-algebra that respectively characterize compatible O-operators and compatible O-operator algebras as Maurer-Cartan elements. Using these characterizations, we define cohomology of these structures and as applications we study formal deformations of compatible O-operators and compatible O-operator algebras. Finally, we consider a brief cohomological study of compatible dendriform algebras and find their relation with the cohomology of compatible associative algebras and compatible O-operators.