Skip to main content

A strong Dixmier-Moeglin equivalence for quantum Schubert cells

Bell, Jason, Launois, Stephane, Nolan, Brendan (2017) A strong Dixmier-Moeglin equivalence for quantum Schubert cells. Journal of Algebra, 487 (1). pp. 269-293. ISSN 0021-8693. (doi:10.1016/j.jalgebra.2017.06.005) (KAR id:62050)

PDF Author's Accepted Manuscript
Language: English


Download (324kB) Preview
[thumbnail of Revised_Final_Version_of_the_SDME_Paper (4) (4).pdf]
Preview
This file may not be suitable for users of assistive technology.
Request an accessible format
Official URL
http://dx.doi.org/10.1016/j.jalgebra.2017.06.005

Abstract

Dixmier and Moeglin gave an algebraic condition and a topological condition for recognising the primitive ideals among the prime ideals of the universal enveloping algebra of a finite-dimensional complex Lie algebra; they showed that the primitive, rational, and locally closed ideals coincide. In modern terminology, they showed that the universal enveloping algebra of a finite-dimensional complex Lie algebra satisfies the Dixmier–Moeglin equivalence.

We define quantities which measure how “close” an arbitrary prime ideal of a noetherian algebra is to being primitive, rational, and locally closed; if every prime ideal is equally “close” to satisfying each of these three properties, then we say that the algebra satisfies the strong Dixmier–Moeglin equivalence . Using the example of the universal enveloping algebra of sl2(C), we show that the strong Dixmier–Moeglin equivalence is strictly stronger than the Dixmier–Moeglin equivalence.

For a simple complex Lie algebra g, a non-root of unity q?0 in an infinite field K, and an element w of the Weyl group of g, De Concini, Kac, and Procesi have constructed a subalgebra Uq[w] of the quantised enveloping K-algebra Uq(g). These quantum Schubert cells are known to satisfy the Dixmier–Moeglin equivalence and we show that they in fact satisfy the strong Dixmier–Moeglin equivalence. Along the way, we show that commutative affine domains, uniparameter quantum tori, and uniparameter quantum affine spaces satisfy the strong Dixmier–Moeglin equivalence.

Item Type: Article
DOI/Identification number: 10.1016/j.jalgebra.2017.06.005
Uncontrolled keywords: Prime ideals; primitive ideals; Dixmier-Moeglin equivalence; quantum Schubert cells
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Depositing User: Stephane Launois
Date Deposited: 13 Jun 2017 08:29 UTC
Last Modified: 16 Feb 2021 13:46 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/62050 (The current URI for this page, for reference purposes)
Launois, Stephane: https://orcid.org/0000-0001-7252-8515
  • Depositors only (login required):

Downloads

Downloads per month over past year