DIXMIER'S PROBLEM 6 FOR SOMEWHAT COMMUTATIVE ALGEBRAS AND DIXMIER'S PROBLEM 3 FOR THE RING OF DIFFERENTIAL OPERATORS ON A SMOOTH IRREDUCIBLE AFFINE CURVE

Abstract

In [6], J. Dixmier posed six problems for the Weyl algebra A1 over a field K of characteristic zero. Problems 3, 5,and 6 were solved respectively by Joseph and Stein [7]; the author [1]; and Joseph [7]. Problems 1, 2, and 4 are still open. For an arbitrary algebra A, Dixmier's problem 6 is essentially aquestion: whether an inner derivation of the algebra A of the type ad f(a), a ∈ A, f(t) ∈ K[t], deg t(f(t)) > 1, has a nonzero eigenvalue. We prove that the answer is negative for many classes of algebras (e.g., rings of differential operators [Formula: see text] on smooth irreducible algebraic varieties, all prime factor algebras of the universal enveloping algebra [Formula: see text] of a completely solvable algebraic Lie algebra [Formula: see text]). This gives an affirmative answer (with a short proof) to an analogue of Dixmier's Problem 6 for certain algebras of small Gelfand–Kirillov dimension, e.g. the ring of differential operators [Formula: see text] on a smooth irreducible affine curve X, Usl(2), etc. (see [3] for details). In this paper an affirmative answer is given to an analogue of Dixmier's Problem 3 but for the ring [Formula: see text].