Rota-Baxter operators and the more general O-operators on Lie algebras together with their interconnected pre-Lie and post-Lie algebras are important algebraic structures with broad applications. 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 cohomology theory of post-Lie algebras is established, with an application to 2-term skeletal operator homotopy post-Lie algebras.

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