Involutive and oriented dendriform algebras

Dendriform algebras are certain splitting of associative algebras and arise naturally from Rota-Baxter operators, shuffle algebras and planar binary trees. In this paper, we first consider involutive dendriform algebras, their cohomology and homotopy analogs. The cohomology of an involutive dendriform algebra splits the Hochschild cohomology of an involutive associative algebra. In the next, we introduce a more general notion of oriented dendriform algebras. We develop a cohomology theory for oriented dendriform algebras that closely related to extensions and governs the simultaneous deformations of dendriform structures and the orientation.