Manin triples on multiplicative Courant algebroids

TL;DR

This paper generalizes the concept of Manin triples to multiplicative Courant algebroids over Lie groupoids, establishing a correspondence with Lie bialgebroid groupoids and providing new insights into related algebraic structures.

Contribution

It extends the characterization of Lie bialgebroids via Manin triples to double structures over Lie groupoids, introducing multiplicative Manin triples and their applications.

Findings

Established a correspondence between Lie bialgebroid groupoids and multiplicative Manin triples.

Provided a new perspective on co-quadratic Lie algebroids and Lie bialgebroid crossed modules.

Connected double structures over Lie groupoids with Dirac structures in Courant algebroids.

Abstract

We extend the characterization of Lie bialgebroids via Manin triples to the context of double structures over Lie groupoids. We consider Lie bialgebroid groupoids, given by LA-groupoids in duality, and establish their correspondence with multiplicative Manin triples, i.e., CA-groupoids equipped with a pair of complementary multiplicative Dirac structures. As an application, we give a new viewpoint to the co-quadratic Lie algebroids of Lang, Qiao, and Yin (2021) and the Manin triple description of Lie bialgebroid crossed modules.