On quantizing semisimple basic algebras

TL;DR

This paper investigates the possibility of polynomial quantization of coordinate rings of certain orbits in semisimple Lie groups, showing some cases admit trivial or consistent quantizations while others do not.

Contribution

It demonstrates the existence of consistent polynomial quantizations for basic nilpotent orbits and proves the impossibility for semisimple orbits in sl(2,R)*.

Findings

Polynomial quantization exists for basic nilpotent orbits.

Any quantization of nilpotent orbits in sl(2,R)* is trivial.

No polynomial quantization exists for basic semisimple orbits in sl(2,R)*.

Abstract

We show that there is a consistent polynomial quantization of the coordinate ring of a basic nilpotent coadjoint orbit of a semisimple Lie group. We also show, at least in the case of a nilpotent orbit in sl(2,R)*, that any such quantization is essentially trivial. Furthermore, we prove that the coordinate ring of a basic semisimple orbit in sl(2,R)* cannot be consistently polynomially quantized.