On quantum flag algebras

TL;DR

This paper extends Kostant's classical result on the quadratic nature of orbit closures in Lie algebra representations to the quantum setting using spectral analysis of braiding operators in quantized universal enveloping algebras.

Contribution

It generalizes Kostant's theorem to quantum groups, employing spectral properties of braiding operators in U_q(g).

Findings

Orbit closures in quantum groups are quadratic cones.

Spectral analysis of braiding operators informs the structure of quantum orbits.

Extension of classical Lie algebra results to quantum algebra setting.

Abstract

Let g be a semisimple Lie algebra over an algebraically closed field k of characteristic 0. Let V be a simple finite-dimensional g-module and let y\in V be a highest weight vector. It is a classical result of B. Kostant that the algebra of functions on the closure of the orbit of y under the simply connected group which corresponds to g is quadratic (i.e. the closuree of the orbit is a quadratic cone). In the present paper we extend this result of Kostant to the case of the quantized universal enveloping algebra U_q(g). The result uses certain information about spectrum of braiding operators for U_q(g) due to Reshetikhin and Drinfeld.