On Lie Bialgebroid Crossed Modules

TL;DR

This paper explores the structure of Lie bialgebroid crossed modules, establishing a correspondence with co-quadratic Manin triples, thereby advancing the understanding of their algebraic and geometric properties.

Contribution

It introduces a one-to-one correspondence between Lie bialgebroid crossed modules and co-quadratic Manin triples, linking algebraic structures with geometric configurations.

Findings

Established a correspondence between Lie bialgebroid crossed modules and co-quadratic Manin triples.

Defined the relationship between Lie algebroid crossed modules and Dirac structures.

Provided a framework connecting algebraic and geometric aspects of Lie bialgebroids.

Abstract

We study Lie bialgebroid crossed modules which are pairs of Lie algebroid crossed modules in duality that canonically give rise to Lie bialgebroids. A one-one correspondence between such Lie bialgebroid crossed modules and co-quadratic Manin triples $(K,P,Q)$ is established, where $K$ is a co-quadratic Lie algebroid and $(P,Q)$ is a pair of transverse Dirac structures in $K$.