Algebra seminar

Brendan Nolan, University of Kent (Canterbury)

“(Generalised) Dixmier-Moeglin Equivalence”

It is typically difficult to classify the irreducible representations of a given algebra A. A good first step is to try to find the primitive ideals i.e. the annihilators of the simple modules. Primitive ideals are prime (in the noncommutative sense) and if one is lucky, the primitive ideals will coincide with two other types of prime ideals: (i) the prime ideals which are locally closed in Zariski topology and (ii) the prime ideals P of A which are such that the centre of the ring of fractions of A/P is an algebraic extension of the ground field (these are called rational ideals). When these ideals coincide, the algebra A is said to satisfy the Dixmier-Moeglin equivalence.

I will outline the proof of the Dixmier-Moeglin equivalence for a well-behaved class of Ore extensions, relying on Cauchon’s Deleting Derivations algorithm. I will discuss the hope that, in some such Ore extensions, the prime ideals which are any given ”distance” from being primitive will coincide with the prime ideals which are the same ”distance” from being locally closed and with the prime ideals which are the same ”distance” from being rational.