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

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.

34 pages

References in corpus (5)

• Anti-pre-Lie algebras, Novikov algebras and commutative 2-cocycles on Lie algebras
• Transposed Poisson structures on Block Lie algebras and superalgebras
• Transposed Poisson structures on Witt type algebras
• The algebraic and geometric classification of transposed Poisson algebras
• Transposed Poisson structures on Galilean and solvable Lie algebras

Cited by in corpus (5)

• Transposed Poisson structures
• Transposed Poisson structures on solvable and perfect Lie algebras
• Transposed Poisson structures on the Lie algebra of upper triangular matrices
• A new approach to the bialgebra theory for relative Poisson algebras
• The algebraic and geometric classification of $\delta$-Novikov algebras