On the Geometry of Cotangent Bundles of Lie Groups

TL;DR

This paper characterizes automorphism groups of cotangent bundles of Lie groups, revealing their super symmetry, exploring their algebraic structures, and analyzing bi-invariant metrics and isometries, with applications to specific Lie groups.

Contribution

It provides a comprehensive analysis of the geometry and algebraic structures of cotangent bundles of Lie groups, including automorphisms, prederivations, and metrics, extending known results.

Findings

Automorphism groups are super symmetric Lie groups.

Prederivations of certain cotangent Lie algebras are all derivations.

The Lie algebra of isometries is generated by inner derivations.

Abstract

Lie groups of automorphisms of cotangent bundles of Lie groups are completely characterized and interesting results are obtained. We give prominence to the fact that the Lie groups of automorphisms of cotangent bundles of Lie groups are super symmetric Lie groups. In the cases of orthogonal Lie lgebras, semi-simple Lie algebras and compact Lie algebras we recover by simple methods interesting co-homological known results. The Lie algebra of prederivations encompasses the one of derivations as a subalgebra. We find out that Lie algebras of cotangent Lie groups (which are not semi-simple) of semi-simple Lie groups have the property that all their prederivations are derivations. This result is an extension of a well known result due to M\"uller. The structure of the Lie algebra of prederivations of Lie algebras of cotangent bundles of Lie groups is explored and we have shown that the Lie algebra of prederivations of Lie algebras of cotangent bundle of Lie groups are reductive Lie algebras. We have studied bi-invariant metrics on cotangent bundles of Lie groups and their isometries. The Lie algebra of the Lie group of isometries of a bi-invariant metric on a Lie group is composed with prederivations of the Lie algebra which are skew-symmetric with respect to the induced orthogonal structure on the Lie algebra. We have shown that the Lie group of isometries of any bi-invariant metric on the cotangent bundle of any semi-simple Lie groups is generated by the exponentials of inner derivations of the Lie cotangent algebra. Last, we have dealt with an introduction to the geometry the Lie group of affine motions of the real line $\mathbb R$, which is a K\"ahlerian Lie group.