Invariant prime ideals in quantizations of nilpotent Lie algebras

TL;DR

This paper classifies invariant prime ideals in certain quantum algebras associated with nilpotent Lie algebras, linking algebraic structures to Weyl group elements and Demazure modules.

Contribution

It explicitly constructs H-invariant prime ideals in quantized algebras and establishes a poset isomorphism with Weyl group orderings, introducing new algebraic and combinatorial insights.

Findings

Explicit construction of H-invariant prime ideals.

Poset of prime ideals is isomorphic to Weyl group order.

Generation of prime ideals using Demazure modules.

Abstract

De Concini, Kac and Procesi defined a family of subalgebras U^w_+ of a quantized universal enveloping algebra U_q(g), associated to the elements of the corresponding Weyl group W. They are deformations of the universal enveloping algebras U(n_+ \cap Ad_w(n_-)) where n_\pm are the nilradicals of a pair of dual Borel subalgebras. Based on results of Gorelik and Joseph and an interpretation of U^w_+ as quantized algebras of functions on Schubert cells, we construct explicitly the H invariant prime ideals of each U^w_+ and show that the corresponding poset is isomorphic to W^{\leq w}, where H is the group of group-like elements of U_q(g). Moreover, for each H-prime of U^w_+ we construct a generating set in terms of Demazure modules related to fundamental representations.