Hermitian structures on cotangent bundles of Lie groups

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

Colonia, Uruguay

Congreso; XVI Coloqio Latinoamericano de Algebra; 2005

Universidad de la República, Uruguay

We study Hermitian structures on the cotangent bundle T*G of a Lie group G which are left invariant with respect to the standard Lie group structure on T*G induced by the coadjoint representation of G on the dual of its Lie algebra. These are in one-to-one correspondence with the left invariant generalized complex structures on G. Using this correspondence, it turns out that when G is four or six dimensional and nilpotent, the cotangent bundle T*G always has a Hermitian structure. However, we prove that if G is a four dimensional solvable (non-nilpotent) Lie group admitting neither complex nor symplectic structures, then G has no left invariant generalized complex structures. Examples of solvable (non-nilpotent) Lie groups G such that the cotangent bundle T*G has Hermitian structures are also given.