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

TL;DR

The paper analyzes Cartan-Schouten metrics, defined as metrics μ that are parallel with respect to the Cartan-Schouten canonical connection, on Lie groups and their cotangent bundles. It proves that for perfect (in particular simple) Lie groups, any Cartan-Schouten metric must be biinvariant, and it derives the resulting structure on T*G, including a full description of biinvariant Cartan-Schouten metrics on T*G when G is simple, distinguishing odd and even dimension cases. A key set of results identifies precise isomorphisms between classical rigid-motion groups and cotangent or tangent bundles of simple Lie groups, notably SE(3) ≅ T*SO(3) ≅ TS O(3) and SE(2,1) ≅ T*SO(2,1), and connects unit dual quaternions and unit dual split quaternions to T*SU(2) and T*SL(2,R) respectively. The work generalizes known SE(3) results to SE(2,1) and T*G for any simple G, establishing biinvariant, symplectic structures and highlighting potential applications to information geometry and related fields via dual quaternions and their geometric interpretations.

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.

Theorems & Definitions (19)

• Definition 2.1
• Definition 2.2
• Proposition 2.1
• Remark 2.1
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Lemma 3.1
• proof