Dixmier’s sixth problem for an algebra \(A\) over a field \(k\) of characteristic zero can be reformulated in the following terms. Let \(a\in A\) and \(f(t)\in k[t]\), \(\deg f(t)>1\). Is it the case that \(\text{ad\,}f(a)\) has a nonzero eigenvalue? Suppose that \({\mathcal D}(X)\) is the ring of differential operators on a smooth irreducible affine algebraic variety. Then all eigenvalues of \(\text{ad\,}f(a)\), \(a\in{\mathcal D}(X)\), vanish.
Let \(A\) be a noncommutative algebra of Gelfand-Kirillov dimension \(<3\) such that \(\overline k\otimes_kA\) is embedded into a division ring in which the centralizer of any noncentral element is commutative. Then for any noncentral element \(a\in A\) each noncentral element \(b\) in \(A\) commuting with \(a\) satisfies one of the conditions: (1) \(b\) is strongly nilpotent, (2) \(b\) is weakly nilpotent, (3) \(b\) is generic with the exception of the set of strongly semi-simple elements, (4) \(b\) is generic with the exception of the set of weakly semi-simple elements, (5) \(b\) is generic.
This theorem is applied to the Weyl algebra \(A_1\), the quantum plane, the ring of differential operators of a smooth irreducible algebraic curve, the universal enveloping algebras \(U(\text{sl}_2)\), \(U_q(\text{sl}_2)\).