Noncommutative Poisson bialgebras

Abstract

In this paper, we introduce the notion of a noncommutative Poisson bialgebra, and establish the equivalence between matched pairs, Manin triples and noncommutative Poisson bialgebras. Using quasi-representations and the corresponding cohomology theory of noncommutative Poisson algebras, we study coboundary noncommutative Poisson bialgebras which leads to the introduction of the Poisson Yang-Baxter equation. A skew-symmetric solution of the Poisson Yang-Baxter equation naturally gives a (coboundary) noncommutative Poisson bialgebra. Rota-Baxter operators, more generally O-operators on noncommutative Poisson algebras, and noncommutative pre-Poisson algebras are introduced, by which we construct skew-symmetric solutions of the Poisson Yang-Baxter equation in some special noncommutative Poisson algebras obtained from these structures.1.1. Noncommutative Poisson algebras and pre-Poisson algebras. The notion of a noncommutative Poisson algebra was first given by Xu in [38], which is especially suitable for geometric situations. 0

Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras

“…On the other hand, there is a bialgebra theory for Poisson algebras, namely, Poisson bialgebras, established in [41]. Note that there is a noncommutative version of Poisson bialgebras given in [34]. Therefore we apply the theory of differential ASI bialgebras to the study of Poisson bialgebras and thus Poisson bialgebras can be constructed from commutative and cocommutative differential ASI bialgebras.…”

Differential antisymmetric infinitesimal bialgebras, coherent derivations and Poisson bialgebras

“…On the other hand, there is a bialgebra theory for Poisson algebras, namely, Poisson bialgebras, established in [36]. Note that there is a noncommutative version of Poisson bialgebras given in [30]. Therefore we apply the theory of differential ASI bialgebras to the study of Poisson bialgebras and thus Poisson bialgebras can be constructed from commutative and cocommutative differential ASI bialgebras.…”

“…These notions and structures are illustrated by the following diagram. The up two layers are notions and structures in commutative and cocommutative differential ASI bialgebras which have been illustrated in the diagram in Section 1.2 and the below two layers are the corresponding notions and structures in Poisson bialgebras given in [36] (also see [30]).…”

Poisson Yang-Baxter equations and $\mathcal{O}$-operators of Poisson superalgebras

“…Finite-dimensional Poisson algebras and Poisson superalgebras over a field F of characteristic zero play important roles in several areas in mathematics and mathematical physics, such as Poisson geometry, integrable systems, non-commutative (algebraic or differential) geometry; see for example [CP94], [Uch08] and [She93]. The Poisson Yang-Baxter equation (PYBE) on a Poisson algebra p has been studied recently in [NB13] and [LBS20] which contains several substantial ramifications, establishing connections between symplectic geometry, PYBEs, and Poisson bialgebras, whereas characterizing specific solutions of the PYBE for a given p is an indispensable and challenging task in terms of the viewpoint of pure mathematics. As a natural generalization of Poisson algebras, Poisson superalgebras contains classes of Lie superalgebras and associative superalgebras.…”