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.

中文翻译：

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对合和定向的树状代数

树状代数是缔合代数的某些分解，并且自然起源于Rota-Baxter算子，混排代数和平面二叉树。在本文中，我们首先考虑对合树突状代数，它们的同调性和同伦类似物。对合的树状代数的同调分解了对合的代数的Hochschild同调。在接下来的内容中，我们将介绍定向树状代数的更一般概念。我们针对定向树枝状代数开发了一种同调理论，该理论与扩展密切相关，并控制树枝状结构和定向的同时变形。