Dixmier's Problem 6 for the Weyl Algebra (The Generic Type Problem)

Abstract

ABSTRACT In Dixmier (1968), the author posed six problems for the Weyl algebra A 1 over a field K of characteristic zero. Problems 3, 6, and 5 were solved respectively by Joseph (1975) and Bavula (2005a). Problems 1, 2, and 4 are still open. In this article a short proof is given to Dixmier's problem 6 for the ring of differential operators 𝒟 (X) on a smooth irreducible algebraic curve X. It is proven that, for a given maximal commutative subalgebra C of 𝒟 (X), (almost) all noncentral elements of it have the same type, more precisely, have exactly one of the following types: (i) strongly nilpotent; (ii) weakly nilpotent; (iii) generic; (iv) generic, except for a subset K*a + K of strongly semi-simple elements; (iv) generic, except for a subset K*a + K of weakly semi-simple elements, where K* := K\\{0}. The same results are true for other popular algebras.