Hermitian structures on cotangent bundles of four dimensional solvable Lie groups

Abstract

We study hermitian structures, with respect to the standard neutral metric on the cotangent bundle $T^{\ast }G$ of a 2n-dimensional Lie group $G$, which are left invariant with respect to the Lie group structure on $T^{\ast }G$ 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 Fernández-Gotay-Gray, it turns out that when $G$ is nilpotent and four or six dimensional, the cotangent bundle $T^{\ast }G$ 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 $T^{\ast }G$ has no hermitian structure or, equivalently, $G$ has no left invariant generalized complex structure.

26 pages. Typos corrected

References in corpus (1)

• New aspects of the ddc-lemma

Cited by in corpus (1)

• Lie bialgebras of complex type and associated Poisson Lie groups