Matched pairs and double construction bialgebras of (transposed) Poisson 3-Lie algebras

TL;DR

This paper develops the theory of double construction bialgebras for Poisson 3-Lie and transposed Poisson 3-Lie algebras, establishing their structures, equivalences, and introducing admissible transposed variants.

Contribution

It introduces double construction bialgebras for Poisson 3-Lie algebras, explores their relations with matched pairs and Manin triples, and defines admissible transposed Poisson 3-Lie algebras.

Findings

Matched pairs, Manin triples, and double construction Poisson 3-Lie bialgebras are equivalent.

Double construction approach does not directly apply to transposed Poisson 3-Lie algebras.

Admissible transposed Poisson 3-Lie algebras are introduced and shown to be equivalent to certain bialgebra structures.

Abstract

Double construction bialgebras for Poisson 3-Lie algebras and transposed Poisson 3-Lie algebras are defined and studied using matched pairs. Poisson 3-Lie algebras and transposed Poisson 3-Lie algebras are constructed on direct sums and tensor products of vector spaces. Matched pairs, Manin triples, and double construction Poisson 3-Lie bialgebras are shown to be equivalent. Since the double construction approach does not apply to bialgebra theory for transposed Poisson 3-Lie algebras, admissible transposed Poisson 3-Lie algebras are introduced. These algebras are both transposed Poisson 3-Lie algebras and Poisson 3-Lie algebras. An equivalence between matched pairs and double construction admissible transposed Poisson 3-Lie bialgebras is established.