\(L_\infty\)-structures and cohomology theory of compatible \(\mathcal{O}\)-operators and compatible dendriform algebras. (English) Zbl 1535.17020
Summary: The notion of \(\mathcal{O}\)-operator is a generalization of the Rota-Baxter operator in the presence of a bimodule over an associative algebra. A compatible \(\mathcal{O}\)-operator is a pair consisting of two \(\mathcal{O}\)-operators satisfying a compatibility relation. A compatible \(\mathcal{O}\)-operator algebra is an algebra together with a bimodule and a compatible \(\mathcal{O}\)-operator. In this paper, we construct a graded Lie algebra and an \(L_\infty\)-algebra that respectively characterize compatible \(\mathcal{O}\)-operators and compatible \(\mathcal{O}\)-operator algebras as Maurer-Cartan elements. Using these characterizations, we define cohomology of these structures and as applications, we study formal deformations of compatible \(\mathcal{O}\)-operators and compatible \(\mathcal{O}\)-operator algebras. Finally, we consider a brief cohomological study of compatible dendriform algebras and find their relationship with the cohomology of compatible associative algebras and compatible \(\mathcal{O}\)-operators.
©2024 American Institute of Physics
©2024 American Institute of Physics
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