A Bijection Between the Adjoint and Coadjoint Orbits of a Semidirect Product

TL;DR

This paper establishes a geometric bijection between adjoint and coadjoint orbits of semidirect product groups, with implications for understanding their structure and homotopy types, exemplified by the affine and Poincaré groups.

Contribution

It proves a general bijection between adjoint and coadjoint orbits for semidirect products under certain conditions, extending the understanding of their geometric and topological properties.

Findings

Bijection exists between adjoint and coadjoint orbits under specific conditions.

Homotopy types of orbits in bijection are equivalent.

Application to affine and Poincaré groups demonstrates the theory's relevance.

Abstract

We prove that there exists a geometric bijection between the sets of adjoint and coadjoint orbits of a semidirect product, provided a similar bijection holds for particular subgroups. We also show that under certain conditions the homotopy types of any two orbits in bijection with each other are the same. We apply our theory to the examples of the affine group and the Poincar\'{e} group, and discuss the limitations and extent of this result to other groups.