Hypersymplectic structures on Courant algebroids

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.