Cohomology and deformations of dendriform algebras, and Dend∞-algebras

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 explicit cohomology of dendriform algebras with coefficients in a representation. Our method avoids the heavy use of operad theory. Deformations of a dendriform algebra A are governed by the cohomology of A with coefficient in itself. Next, we study Dend∞-algebras (dendriform algebras up to homotopy) in which the dendriform identities hold up to certain homotopy. They are a certain splitting of A∞-algebras. We define Rota-Baxter operator on A∞-algebras which naturally gives rise to Dend∞-algebras. Finally, we classify skeletal and strict Dend∞-algebras.