Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota–Baxter Operators

Abstract

We will introduce an associative (or quantum) version of Poisson structure tensors. This object is defined as an operator satisfying a “generalized” Rota–Baxter identity of weight zero. Such operators are called generalized Rota–Baxter operators. We will show that generalized Rota–Baxter operators are characterized by a cocycle condition so that Poisson structures are so. By analogy with twisted Poisson structures, we propose a new operator “twisted Rota–Baxter operators,” which is a natural generalization of generalized Rota–Baxter operators. It is known that classical Rota–Baxter operators are closely related with dendriform algebras. We will show that twisted Rota–Baxter operators induce NS-algebra, which is a twisted version of dendriform algebra. The twisted Poisson condition is considered as a Maurer–Cartan equation up to homotopy. We will show the twisted Rota–Baxter condition also is so. And we will study a Poisson-geometric reason, how the twisted Rota–Baxter condition arises.