Hypersymplectic structures on Courant algebroids

TL;DR

This paper introduces hypersymplectic structures on Courant algebroids, establishing a correspondence with hyperk"ahler structures, and explores their applications to various Lie algebroid cases.

Contribution

It defines hypersymplectic structures on Courant algebroids and links them to hyperk"ahler structures, simplifying their characterization and encompassing structures with torsion.

Findings

Established a one-to-one correspondence between hypersymplectic and hyperk"ahler structures.

Extended the framework to include structures with torsion on Lie algebroids.

Analyzed hypersymplectic structures on doubles of Lie, quasi-Lie, and proto-Lie bialgebroids.

Abstract

We introduce the notion of hypersymplectic structure on a Courant algebroid and we prove the existence of a one-to-one correspondence between hypersymplectic and hyperk\"ahler structures. This correspondence provides a simpler way to define a hyperk\"ahler structure on a Courant algebroid. We show that hypersymplectic structures on Courant algebroids encompass hyperk\"ahler and hyperk\"ahler structures with torsion on Lie algebroids. In the latter, the torsion existing at the Lie algebroid level is incorporated in the Courant structure. Cases of hypersymplectic structures on Courant algebroids which are doubles of Lie, quasi-Lie and proto-Lie bialgebroids are investigated.