A bialgebra theory for transposed Poisson algebras via anti-pre-Lie bialgebras and anti-pre-Lie-Poisson bialgebras

TL;DR

This paper develops a new bialgebra theory for transposed Poisson algebras using anti-pre-Lie structures and Manin triples with respect to commutative 2-cocycles, extending the algebraic framework beyond traditional methods.

Contribution

It introduces anti-pre-Lie bialgebras and anti-pre-Lie-Poisson bialgebras, providing a novel approach to the bialgebra theory for transposed Poisson algebras via Manin triples.

Findings

Defined anti-pre-Lie bialgebras as Manin triples of Lie algebras with commutative 2-cocycles.

Characterized anti-pre-Lie-Poisson bialgebras through Manin triples respecting invariant bilinear forms.

Explored coboundary cases, classical Yang-Baxter equations, and $ ext{O}$-operators related to these structures.

Abstract

The approach for Poisson bialgebras characterized by Manin triples with respect to the invariant bilinear forms on both the commutative associative algebras and the Lie algebras is not available for giving a bialgebra theory for transposed Poisson algebras. Alternatively, we consider Manin triples with respect to the commutative 2-cocycles on the Lie algebras instead. Explicitly, we first introduce the notion of anti-pre-Lie bialgebras as the equivalent structure of Manin triples of Lie algebras with respect to the commutative 2-cocycles. Then we introduce the notion of anti-pre-Lie Poisson bialgebras, characterized by Manin triples of transposed Poisson algebras with respect to the bilinear forms which are invariant on the commutative associative algebras and commutative 2-cocycles on the Lie algebras, giving a bialgebra theory for transposed Poisson algebras. Finally the coboundary cases and the related structures such as analogues of the classical Yang-Baxter equation and $\mathcal O$-operators are studied.