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

Kyousuke Uchino
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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-algebras 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.

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Kyousuke Uchino. 2008-07-24. Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota-Baxter Operators. https://doi.org/10.1007/s11005-008-0259-2

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