Involutive and oriented dendriform algebras

TL;DR

This paper introduces involutive and oriented dendriform algebras, exploring their cohomology, homotopy, and deformation theories, thereby expanding the algebraic framework related to associative structures and Rota-Baxter operators.

Contribution

It develops the cohomology and deformation theories for involutive and oriented dendriform algebras, extending the understanding of their algebraic and homotopical properties.

Findings

Cohomology of involutive dendriform algebras splits Hochschild cohomology.

Introduces a cohomology theory for oriented dendriform algebras.

Establishes deformation theories for these algebraic structures.

Abstract

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.