# 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)

## 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 10.1016/j.jalgebra.2017.06.005 Prime ideals; primitive ideals; Dixmier-Moeglin equivalence; quantum Schubert cells Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science Stephane Launois 13 Jun 2017 08:29 UTC 16 Feb 2021 13:46 UTC https://kar.kent.ac.uk/id/eprint/62050 (The current URI for this page, for reference purposes) https://orcid.org/0000-0001-7252-8515