Invariant quantization in one and two parameters on semisimple coadjoint orbits of simple Lie groups

TL;DR

This paper classifies and constructs invariant quantizations of functions on semisimple coadjoint orbits of simple Lie groups, focusing on one and two parameter cases with applications to compatible Poisson brackets.

Contribution

It provides a classification of invariant brackets on semisimple orbits and constructs explicit two-parameter quantizations where compatible Poisson brackets exist.

Findings

The Poisson bracket must be the sum of an R-matrix and an invariant bracket.

The family of such brackets has dimension equal to the rank of the orbit.

Not all semisimple orbits admit compatible pairs of Poisson brackets.

Abstract

We study one and two parameter quantizations of the function algebra on a semisimple orbit in the coadjoint representation of a simple Lie group subject to the condition that the multiplication on the quantized algebra is invariant under action of the Drinfeld-Jimbo quantum group. We prove that the corresponding Poisson bracket must be the sum of the so-called R-matrix bracket and an invariant bracket. We classify such brackets for all semisimple orbits and show that they form a family of dimension equal to the rank equal to the second cohomology group of the orbit and then we quantize these brackets. A two parameter (or double) quantization corresponds to a pair of compatible Poisson brackets: the first is as described above and the second is the Kirillov-Kostant-Souriau bracket on the orbit. Not all semisimple orbits admit a compatible pair of Poisson brackets. We classify the semisimple orbits for which such pairs exist and construct the corresponding two parameter quantization of these pairs in some of the cases.