On dual quaternions, dual split quaternions and Cartan-Schouten metrics on perfect Lie groups

TL;DR

This paper explores Cartan-Schouten metrics on perfect Lie groups, revealing their biinvariance, computing such metrics on cotangent bundles of simple Lie groups, and establishing isomorphisms between certain dual quaternion groups and tangent or cotangent bundles.

Contribution

It demonstrates that Cartan-Schouten metrics on perfect Lie groups are necessarily biinvariant and computes these metrics on cotangent bundles, also establishing key isomorphisms with dual quaternion groups.

Findings

Cartan-Schouten metrics on perfect Lie groups are biinvariant.

Explicit computation of Cartan-Schouten metrics on cotangent bundles of simple Lie groups.

Isomorphisms between dual quaternion groups and tangent or cotangent bundles of specific Lie groups.

Abstract

We discuss Cartan-Schouten metrics (Riemannian or pseudo-Riemannian metrics that are parallel with respect to the Cartan-Schouten canonical connection) on perfect Lie groups. Applications are foreseen in Information Geometry. Throughout this work, the tangent bundle TG and the cotangent bundle T*G of a Lie group G, are always endowed with their Lie group structures induced by the right trivialization. We show that TG and T*G are isomorphic if G possesses a biinvariant Riemannian or pseudo-Riemannian metric. We also show that, if on a perfect Lie group, there exists a Cartan-Schouten metric, then it must be biinvariant. We compute all such metrics on the cotangent bundles of simple Lie groups. We further show the following. Endowed with their canonical Lie group structures, the set of unit dual quaternions is isomorphic to TSU(2), the set of unit dual split quaternions is isomorphic to T*SL(2,R). The group SE(3) of special rigid displacements of the Euclidean 3-space is isomorphic to T*SO(3). The group SE(2,1) of special rigid displacements of the Minkowski 3-space is isomorphic to T*SO(2,1). Some results on SE(3) by N. Miolane and X. Pennec, and M. Zefran, V. Kumar and C. Croke, are generalized to SE(2,1) and to T*G, for any simple Lie group G.