Automorphisms of cotangent bundles of Lie groups

TL;DR

This paper characterizes the automorphisms and derivations of the Lie algebra of the cotangent bundle of a Lie group, revealing structural insights and connections to bi-invariant metrics and operators.

Contribution

It provides a complete characterization of derivations of the Lie algebra of the cotangent bundle of a Lie group and explores related operator spaces and invariance properties.

Findings

Full characterization of derivations of the Lie algebra of T*G

Identification of bi-invariant tensor spaces on G

Isomorphism between certain operator spaces and invariant forms

Abstract

Let G be a Lie group, $T^*G$ 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 $T^*G$. More precisely, amongst other results, we fully characterize the space of all derivations of the Lie algebra of $T^*G$. 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.