Dixmier and Moeglin showed that if L is a finite-dimensional complex Lie algebra then the primitive ideals of the enveloping algebra U(L) are the prime ideals of Spec(U(L)) that are locally closed in the Zariski topology. In addition, they proved that a prime ideal P of U(L) is primitive if and only if the Goldie ring of quotients of U(L)/P has the property that its centre is just the base field of the complex numbers. Algebras that share this characterization of primitive ideals are said to satisfy the Dixmier-Moeglin equivalence. We give an overview of this property and mention some recent work on proving this equivalence holds for certain classes of twisted homogenous coordinate rings and classes of Hopf algebras of small Gelfand-Kirillov dimension.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
