A strong Dixmier-Moeglin equivalence for quantum Schubert cells

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 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 sl_2(C), we show that the strong Dixmier-Moeglin equivalence is stronger than the Dixmier-Moeglin equivalence. For a simple complex Lie algebra g, a non root of unity q\neq 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 U_q[w] of the quantised enveloping K-algebra U_q(g). These quantum Schubert cells U_q[w] 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.