Year: 2026
Type: preprint
Abstract: Poisson algebras are the quasiclassical limits of associative algebra deformations of commutative associative algebras. This paper extends this process to the level of bialgebras. We derive Poisson bialgebras as the quasiclassical limits of antisymmetric infinitesimal bialgebra deformations of commutative and cocommutative antisymmetric infinitesimal bialgebras. It might be regarded as the ``infinitesimal" version of the quantization process of Lie bialgebras in terms of Hopf algebras. Such deformations-quasiclassical limits process for a Poisson bialgebra is equivalently characterized in terms of the introduced notions of deformations of a matched pair of associative algebras as well as a standard Manin triple of associative algebras, whose corresponding quasiclassical limits are a matched pair of Poisson algebras and a standard Manin triple of Poisson algebras, respectively. We illustrate these equivalent deformations-quasiclassical limits processes via coherent derivations.
Source: arXiv (Cornell University)
Authors: Chen, Siyuan, Bai, Chengming
Institutions: Hunan Institute of Science and Technology, Nankai University
Language: English
Topic: Advanced Topics in Algebra
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G896127649
OpenAlex ID: W7204886831