Linear phase space deformations with angular momentum symmetry

TL;DR

This paper classifies linear deformations of standard phase space that preserve angular momentum symmetry, revealing their structure as degenerations of coadjoint orbits related to Grassmannians.

Contribution

It provides a classification of phase space deformations maintaining $ ext{O}(n)$ symmetry and links standard phase spaces to coadjoint orbits and Grassmannians.

Findings

Standard phase space can be viewed as a degeneration of coadjoint orbits.

Deformations preserve symplectic structure and angular momentum symmetry.

Connections established between phase space, Grassmannians, and coadjoint orbits.

Abstract

Motivated by the work of Leznov--Mostovoy, we classify the linear deformations of standard $2n$-dimensional phase space that preserve the obvious symplectic $\mathfrak{o}(n)$-symmetry. As a consequence, we describe standard phase space, as well as $T^{*}S^{n}$ and $T^{*}\mathbb{H}^{n}$ with their standard symplectic forms, as degenerations of a 3-dimensional family of coadjoint orbits, which in a generic regime are identified with the Grassmannian of oriented 2-planes in $\mathbb{R}^{n+2}$.