Admissible Poisson bialgebras
Jinting Liang, Jiefeng Liu, Chengming Bai

TL;DR
This paper develops a systematic bialgebra theory for admissible Poisson algebras, introducing new concepts like adm-Poisson Yang-Baxter equations and $ ext{O}$-operators, simplifying the study of Poisson bialgebras.
Contribution
It introduces adm-Poisson bialgebras, establishes their relation to Manin triples, and connects Yang-Baxter equations with adm-Poisson structures, offering a unified framework with a single operation.
Findings
Established a bialgebra theory for adm-Poisson algebras.
Connected adm-Poisson Yang-Baxter equation with associative Yang-Baxter equation.
Constructed adm-Poisson bialgebras via $ ext{O}$-operators and pre-adm-Poisson algebras.
Abstract
An admissible Poisson algebra (or briefly, an adm-Poisson algebra) gives an equivalent presentation with only one operation for a Poisson algebra. We establish a bialgebra theory for adm-Poisson algebras independently and systematically, including but beyond the corresponding results on Poisson bialgebras given in [27]. Explicitly, we introduce the notion of adm-Poisson bialgebras which are equivalent to Manin triples of adm-Poisson algebras as well as Poisson bialgebras. The direct correspondence between adm-Poisson bialgebras with one comultiplication and Poisson bialgebras with one cocommutative and one anti-cocommutative comultiplications generalizes and illustrates the polarization-depolarization process in the context of bialgebras. The study of a special class of adm-Poisson bialgebras which include the known coboundary Poisson bialgebras in [27] as a proper subclass in general,…
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology
