Hermitian structures on cotangent bundles of four dimensional solvable Lie groups

TL;DR

This paper investigates hermitian structures on cotangent bundles of four-dimensional solvable Lie groups, establishing conditions for their existence and linking them to generalized complex structures, with specific results for nilpotent and non-nilpotent cases.

Contribution

It characterizes when cotangent bundles of four-dimensional solvable Lie groups admit hermitian structures, connecting them to generalized complex structures and providing new existence and non-existence results.

Findings

Cotangent bundles of nilpotent 4- and 6-dimensional Lie groups always admit hermitian structures.

Four-dimensional solvable Lie groups without complex or symplectic structures have cotangent bundles that do not admit hermitian structures.

The correspondence between hermitian structures and generalized complex structures is established and utilized.

Abstract

We study hermitian structures, with respect to the standard neutral metric on the cotangent bundle $T^*G$ of a 2n-dimensional Lie group $G$, which are left invariant with respect to the Lie group structure on $T^*G$ induced by the coadjoint action. These are in one-to-one correspondence with left invariant generalized complex structures on $G$. Using this correspondence and results of Cavalcanti-Gualtieri and Fern\'andez-Gotay-Gray, it turns out that when $G$ is nilpotent and four or six dimensional, the cotangent bundle $T^*G$ always has a hermitian structure. However, we prove that if $G$ is a four dimensional solvable Lie group admitting neither complex nor symplectic structures, then $T^*G$ has no hermitian structure or, equivalently, $G$ has no left invariant generalized complex structure.