Isotropic embeddings of coadjoint orbits and magnetic geodesic flows

TL;DR

This paper explores isotropic embeddings of coadjoint orbits for compact Lie groups, extending known results for SU(n) to SO and Sp, and applies these to relate magnetic geodesic flows to spin chain systems.

Contribution

It introduces new isotropic embedding techniques for SO and Sp coadjoint orbits and connects magnetic geodesic flows with classical spin chains.

Findings

Established isotropic embeddings for SO and Sp coadjoint orbits.

Proved equivalence between magnetic geodesic flows and certain spin chain models.

Abstract

We consider isotropic and Lagrangian embeddings of coadjoint orbits of compact Lie groups into products of coadjoint orbits. After reviewing the known facts in the case of $\mathrm{SU}(n)$ we initiate a similar study for $\mathrm{SO}$ and $\mathrm{Sp}$ cases. In the second part we apply this to the study of dynamical systems with $\mathrm{SU}(n)$ symmetry, proving equivalence between systems of two types: those describing magnetic geodesic flow on flag manifolds and classical `spin chains' of a special type.