Matching relative Rota-Baxter algebras, matching dendriform algebras and their cohomologies

Abstract

The notion of matching Rota-Baxter algebras was recently introduced by Gao, Guo, and Zhang motivated by the study of algebraic renormalization of regularity structures. In this paper, we introduce matching relative Rota-Baxter algebras that are closely related to matching dendriform algebras. We define the cohomology of a matching relative Rota-Baxter algebra using the classical Hochschild cohomology and a new cohomology induced by the matching operators. As an application, we show that our cohomology governs the formal deformation theory of the matching relative Rota-Baxter algebra. Finally, we define the cohomology of a matching dendriform algebra and study homotopy matching dendriform algebras.