A proof of the Goodearl-Lenagan polynormality conjecture

TL;DR

This paper proves the Goodearl-Lenagan polynormality conjecture for quantum nilpotent algebras, showing all torus invariant prime ideals are polynormal and providing explicit generators, with implications for algebraic structure and separation properties.

Contribution

It provides an explicit constructive proof that all torus invariant prime ideals in quantum nilpotent algebras are polynormal, confirming the conjecture and analyzing their structural properties.

Findings

All torus invariant prime ideals are polynormal.

Explicit polynormal generating sets for quantum matrices.

Spec U^w_-(g) is normally separated and algebras are catenary.

Abstract

The quantum nilpotent algebras U^w_-(g), defined by De Concini-Kac-Procesi and Lusztig, are large classes of iterated skew polynomial rings with rich ring theoretic structure. In this paper, we prove in an explicit way that all torus invariant prime ideals of the algebras U^w_-(g) are polynormal. In the special case of the algebras of quantum matrices, this construction yields explicit polynormal generating sets consisting of quantum minors for all of their torus invariant prime ideals. This gives a constructive proof of the Goodearl-Lenagan polynormality conjecture. Furthermore we prove that Spec U^w_-(g) is normally separated for all simple Lie algebras g and Weyl group elements w, and deduce from it that all algebras U^w_-(g) are catenary.