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-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.

18 pages. Final version

References in corpus (5)

• Loday-type Algebras and the Rota-Baxter Relation
• Quantum Bi-Hamiltonian Systems
• Integrable Renormalization II: the general case
• Spitzer's Identity and the Algebraic Birkhoff Decomposition in pQFT
• Integrable Renormalization I: the Ladder Case

Cited by in corpus (14)

• Splitting of operations, Manin products and Rota-Baxter operators
• Some results on L-dendriform algebras
• Twisted Rota-Baxter operators and Reynolds operators on Lie algebras and NS-Lie algebras
• Factorizable Lie bialgebras, quadratic Rota-Baxter Lie algebras and Rota-Baxter Lie bialgebras
• Cohomology and deformations of weighted Rota-Baxter operators
• The $L_{\infty }$-deformations of associative Rota-Baxter algebras and homotopy Rota-Baxter operators
• Totally compatible associative and Lie dialgebras, tridendriform algebras and PostLie algebras
• Kupershmidt-(dual-)Nijenhuis structures on a Lie algebra with a representation
• Extensions and automorphisms of Rota-Baxter groups
• Anti-dendriform algebras, new splitting of operations and Novikov type algebras
• $\mathcal{O}$-operators and related structures on Leibniz algebras
• Rota--Baxter operators and skew left brace structures over Heisenberg Group
• Cohomology and relative Rota-Baxter-Nijenhuis structures on LieYRep pairs
• Fundamental Theorems of Calculus and Zinbiel Algebras