Hermitian structures on cotangent bundles of solvable Lie groups

L.C. DE ANDRÉS, M. L. BARBERIS, I. DOTTI, M. FERNÁNDEZ

Campinas, Brasil

Congreso; I Congreso Latinoamericano de Grupos de Lie en Geometría; 2006

Unicamp, Brasil

We study  hermitian structures, with respect to the standard neutralmetric on the cotangent bundle $T^*G$ of a 2n-dimensional Lie group$G$, which are left invariant with respect to the Lie groupstructure on $T^*G$ induced by the coadjoint action. These are inone-to-one correspondence with  left invariant generalized complexstructures on $G$. Using this correspondence and results ofCavalcanti-Gualtieri and Fernández-Gotay-Gray, it turns out that when $G$ is nilpotentand four or six dimensional, the cotangent bundle $T^*G$ always hasa hermitian structure. However, we prove that if $G$ is a fourdimensional  solvable Lie group admitting neither complex norsymplectic structures, then $T^*G$ has no hermitian structure or,equivalently, $G$ has no left invariant generalized complexstructure.