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

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.

Key Words:

• Centralizer
• The sixth problem of Dixmier
• The Weyl algebra
• Type of element

Mathematics Subject Classification 2000:

• 16S32
• 13N15
• 13N10
• 16D30

ACKNOWLEDGMENT

The author would like to thank J. Dixmier for comments on his problems from Dixmier (1968), and D. Jordan and T. Lenagan for pointing out on the article of Goodearl (1983).

Notes

Communicated by T. Lenagan.