Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota–Baxter Operators
Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota–Baxter Operators
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DOI：
10.1007/s11005-008-0259-2
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发表时间：
2007-01
影响因子：
1.2
通讯作者：
K. Uchino
中科院分区：
文献类型：
--
作者：
K. Uchino
我们将介绍泊松结构张量的结合（或量子）版本。这个对象被定义为满足“广义”Rota-Baxter恒等式的权重为零的算子。这种算子称为广义Rota-Baxter算子。我们将证明广义Rota-Baxter算子的特征在于一个上循环条件，使泊松结构。通过类比扭Poisson结构，我们提出了一种新的算子“扭Rota-Baxter算子”，它是广义Rota-Baxter算子的自然推广.经典的Rota-Baxter算子与树状代数密切相关。我们将证明扭曲Rota-Baxter算子诱导NS-代数，它是树状代数的扭曲版本。扭泊松条件被认为是一个Maurer-Cartan方程的同伦。我们将证明扭曲的Rota-Baxter条件也是如此。我们将研究一个泊松几何原因，扭曲Rota-Baxter条件如何产生。
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.