Extending structures for pre-Poisson algebras and pre-Poisson bialgebras
Qianwen Zhu, Guilai Liu, Qinxiu Sun
TL;DR
This work develops a comprehensive extending structures framework for pre-Poisson algebras by employing unified products to classify all possible enrichments of a given subalgebra. It introduces $H^2_p(V,A)$ as a cohomological classifier and establishes bijections to Extd$(E,A)$ and Extd'(E,A)$, thereby reducing ES-problems to computational invariants. The paper then analyzes special cases via crossed and bicrossed products, flag data for one-dimensional extensions, and matched pairs, linking these constructions to the factorization problem and to explicit classifications. In the bialgebraic part, it shows that pre-Poisson bialgebras are equivalent to certain matched pairs, develops the PPYBE, and characterizes coboundary and quadratic structures through dualities with Zinbiel and pre-Lie theories. Collectively, the results provide a unified algebraic toolkit for constructing and classifying pre-Poisson (bi)algebras and connect extending structures with Yang–Baxter-type equations, enriching both the theory and potential applications in Poisson-type settings.
Abstract
In this paper, we explore the extending structures problem by the unified product for pre-Poisson algebras. In particular, the crossed product and the factorization problem are investigated. Furthermore, a special case of extending structures is studied under the case of pre-Poisson algebras, which leads to the discussion of bicrossed products and matched pairs of pre-Poisson algebras. We develop a bialgebra theory for pre-Poisson algebras and establish the equivalence between matched pairs and pre-Poisson bialgebras. We study coboundary pre-Poisson bialgebras, which lead to the introduction of the pre-Poisson Yang-Baxter equation (PPYBE). A symmetric solution of the PPYBE naturally gives a coboundary pre-Poisson bialgebra.