Automorphisms of cotangent bundles of Lie groups

Abstract

Let G be a Lie group, $T^{\ast }G$ its cotangent bundle with its natural Lie group structure obtained by performing a left trivialization of T^*G and endowing the resulting trivial bundle with the semi-direct product, using the coadjoint action of G on the dual space of its Lie algebra. We investigate the group of automorphisms of the Lie algebra of $T^{\ast }G$. More precisely, amongst other results, we fully characterize the space of all derivations of the Lie algebra of $T^{\ast }G$. As a byproduct, we also characterize some spaces of operators on G amongst which, the space J of bi-invariant tensors on G and prove that if G has a bi-invariant Riemannian or pseudo-Riemannian metric, then J is isomorphic to the space of linear maps from the Lie algebra of G to its dual space which are equivariant with respect to the adjoint and coadjoint actions, as well as that of bi-invariant bilinear forms on G. We discuss some open problems and possible applications.

V2: 27 pages, Latex, a few minor results added and the paper layout reorganised. The last version appeared at Afr. Diaspora J. Math

References in corpus (2)

• A hyperkahler structure on the cotangent bundle of a complex Lie group
• Poisson structures on the cotangent bundle of a Lie group or a principal bundle and their reductions