Poisson bialgebras

Abstract

We introduce a notion of Poisson bialgebra as an analogue of a Lie bialgebra of Drinfeld. Poisson bialgebras exhibit many familiar properties of Lie bialgebras. In particular, they can be constructed from a combination of the classical Yang-Baxter equation and the associative Yang-Baxter equation and there exists a natural Drinfeld classical double. Moreover, Poisson bialgebras are related to certain algebraic structures and they fit naturally into a framework to construct compatible Poisson brackets in integrable systems.

Constructing Hom-Poisson color algebras

“…vanishe by (6) and the ε-skew-symmetry, c 1 +c 2 +c 3 , f 1 +f 2 +f 3 and e 1 +e 2 +e 3 vanishe by (5) and the ε-skew-symmetry, g 1 + g 2 + g 3 vanishes by (3). It follows that x,y,z ε(z, x){α(x), {y, z}} = 0.…”

“…Using axiom (6) and ε-Hom-Jacobi identity, it is immediate to see that (6) holds. Now, we prove (5) for · and [−, −] .…”

“…Among other results, the authors proved in [5] that tridendriform and Rota-Baxter Lie algebras lead to the structures of post-Lie algebras. Post-Poisson algebras [6] are algebraic structures containing post-Lie algebras [4], [9] and pre-Poisson algebras [13] which contain pre-Lie algebras (or left-symmetric algebras) and Zinbiel algebras, both structures being connected by some compatibility conditions.Classical (or ordinary) algebras were extended to generalized (or color or graded) algebras, among which one can cite : Simple Jordan color algebras arising from associative graded algebras [11], Representations and cocycle twists of color Lie algebras [16], On the classification of 3-dimensional coloured Lie algebras [15]. These structures are well-known to physicists and to mathematicans studying differential geometry and homotopy theory.…”

“…Among other results, the authors proved in [5] that tridendriform and Rota-Baxter Lie algebras lead to the structures of post-Lie algebras. Post-Poisson algebras [6] are algebraic structures containing post-Lie algebras [4], [9] and pre-Poisson algebras [13] which contain pre-Lie algebras (or left-symmetric algebras) and Zinbiel algebras, both structures being connected by some compatibility conditions.…”

Noncommutative Poisson bialgebras

“…Note that the usual Poisson algebras are coherent. Thus, the bialgebra theory for noncommutative Poisson algebras established in this paper contains all the results of Poisson bialgebras given in [32].…”