Homotopy Rota–Baxter operators and post-Lie algebras

Rota–Baxter operators and the more general \mathcal{O} -operators, together with their interconnected 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 quantum field theory. This paper introduces the notions of a homotopy Rota–Baxter operator and a homotopy \mathcal{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 \mathcal{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 cohomology theory of post-Lie algebras is established, with an application to 2-term skeletal operator homotopy post-Lie algebras.