Homotopy Rota-Baxter operators and post-Lie algebras

Rota-Baxter operators and the more general O-operators, together with their intercon-nected pre-Lie and post-Lie algebras, are important algebraic structures, with Rota-Baxter operators and pre-Lie algebras instrumental in the Connes-Kreimer approach to renormalization of quan-tum field theory. This paper introduces the notions of a homotopy Rota-Baxter operator and a homotopy O-operator on a symmetric graded Lie algebra. Their characterization by Maurer-Cartan elements of suitable differential graded Lie algebras is provided. Through the action of a homotopy O-operator on a symmetric graded Lie algebra, we arrive at the notion of an operator homotopy post-Lie algebra, together with its characterization in terms of Maurer-Cartan elements. A coho-mology theory of post-Lie algebras is established, with an application to 2-term skeletal operator homotopy post-Lie algebras.