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.
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Post doctoral student supported by Japan Society for the Promotion of Science.
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Uchino, K. Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota–Baxter Operators. Lett Math Phys 85, 91–109 (2008). https://doi.org/10.1007/s11005-008-0259-2
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DOI: https://doi.org/10.1007/s11005-008-0259-2