Hermitian structures on cotangent bundles of four dimensional solvable Lie groups

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

OSAKA JOURNAL OF MATHEMATICS

Osaka University

Lugar: Osaka; Año: 2007 vol. 44 p. 765 - 793

0030-6126

Abstract.We study  hermitian structures, with respect to the standardneutral metric on the cotangent bundle $T^*G$ of a 2n-dimensionalLie group $G$, which are left invariant with respect to the Liegroup structure on $T^*G$ induced by the coadjoint action. Theseare in one-to-one correspondence with  left invariant generalizedcomplex structures on $G$. Using this correspondence and resultsof Cavalcanti-Gualtieri and Fernández-Gotay-Gray, it turns out that when $G$ isnilpotent and four or six dimensional, the cotangent bundle $T^*G$always has a hermitian structure. However, we prove that if $G$ isa four dimensional  solvableLie group admitting neither complexnor symplectic structures, then $T^*G$ has no hermitian structure or,equivalently, $G$ has no left invariant generalized complex structure.