Strata of prime ideals of De Concini-Kac-Procesi algebras and Poisson geometry

TL;DR

This paper explores the structure of prime ideals in De Concini-Kac-Procesi algebras and their relation to Poisson geometry, providing formulas for the dimensions of associated strata and rational Casimir fields.

Contribution

It derives explicit formulas for the dimensions of Goodearl-Letzter strata and the transcendence degree of Casimir fields on Richardson varieties in the context of these algebras.

Findings

Formulas for the dimensions of strata

Calculation of transcendence degree of Casimir fields

Connection between prime ideals and Poisson geometry

Abstract

To each simple Lie algebra g and an element w of the corresponding Weyl group De Concini, Kac and Procesi associated a subalgebra U^w_- of the quantized universal enveloping algebra U_q(g), which is a deformation of the universal enveloping algebra U(n_- \cap w(n_+)) and a quantization of the coordinate ring of the Schubert cell corresponding to w. The torus invariant prime ideals of these algebras were classified by M\'eriaux and Cauchon [25], and the author [30]. These ideals were also explicitly described in [30]. They index the the Goodearl-Letzter strata of the stratification of the spectra of U^w_- into tori. In this paper we derive a formula for the dimensions of these strata and the transcendence degree of the field of rational Casimirs on any open Richardson variety with respect to the standard Poisson structure [15].