Hypercomplex structures on Courant algebroids

TL;DR

This paper explores hypercomplex structures on Courant algebroids, establishing their equivalence through two different characterizations involving Nijenhuis concomitants and torsionfree connections, thereby unifying various geometric structures.

Contribution

It proves the equivalence of two characterizations of hypercomplex structures on Courant algebroids, advancing the understanding of their geometric properties.

Findings

Equivalence of two characterizations of hypercomplex structures

Connection between Nijenhuis concomitants and torsionfree connections

Unification of holomorphic symplectic and hypercomplex structures

Abstract

Hypercomplex structures on Courant algebroids unify holomorphic symplectic structures and usual hypercomplex structures. In this note, we prove the equivalence of two characterizations of hypercomplex structures on Courant algebroids, one in terms of Nijenhuis concomitants and the other in terms of (almost) torsionfree connections for which each of the three complex structures is parallel.