$L_\infty$-structures and cohomology theory of compatible $\mathcal {O}$-operators and compatible dendriform algebras

TL;DR

This paper develops a cohomology theory for compatible $ ext{O}$-operators and dendriform algebras using $L_$-algebra structures, enabling the study of their deformations and interrelations.

Contribution

It constructs $L_$-algebra frameworks characterizing compatible $ ext{O}$-operators and algebras, introducing their cohomology and deformation theories, and explores connections with dendriform and associative algebras.

Findings

Defined cohomology for compatible $ ext{O}$-operators and algebras.

Established $L_$-algebra structures for these objects.

Analyzed deformations and relationships with dendriform and associative algebras.

Abstract

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.