Cohomology and deformations of dendriform algebras, and $\mathrm{Dend}_\infty$-algebras

TL;DR

This paper develops a cohomology theory for dendriform algebras, explores their deformations and homotopy versions called $ ext{Dend}_$-algebras, and classifies special cases, linking them to $A_$-algebras via Rota-Baxter operators.

Contribution

It introduces cohomology for dendriform algebras, defines $ ext{Dend}_$-algebras up to homotopy, and connects these structures through Rota-Baxter operators, providing classification results.

Findings

Cohomology of dendriform algebras is established.

Deformations governed by the cohomology are characterized.

Classification of skeletal and strict $ ext{Dend}_$-algebras is achieved.

Abstract

A dendriform algebra is an associative algebra whose product splits into two binary operations and the associativity splits into three new identities. These algebras arise naturally from some combinatorial objects and through Rota-Baxter operators. In this paper, we start by defining cohomology of dendriform algebras with coefficient in a representation. The deformation of a dendriform algebra $A$ is governed by the cohomology of $A$ with coefficient in itself. Next we study $\mathrm{Dend}_\infty$-algebras (dendriform algebras up to homotopy) in which the dendriform identities hold up to certain homotopy. They are certain splitting of $A_\infty$-algebras. We define Rota-Baxter operator on $A_\infty$-algebras which naturally gives rise to $\mathrm{Dend}_\infty$-algebras. Finally, we classify skeletal and strict $\mathrm{Dend}_\infty$-algebras.