On Dixmier’s Fourth Problem

Abstract

Let $\mathfrak {g}$ be a finite dimensional Lie algebra over an algebraically closed field k of characteristic zero. Denote by $U(\mathfrak {g})$ its enveloping algebra with quotient division ring $D(\mathfrak {g})$ . In 1974, at the end of his book “Algebres enveloppantes”, Jacques Dixmier listed 40 open problems, of which the fourth one asked if the center $Z(D(\mathfrak {g}))$ is always a purely transcendental extension of k. We show this is the case if $\mathfrak {g}$ is algebraic whose Poisson semi-center $Sy(\mathfrak {g})$ is a polynomial algebra over k. This can be applied to many biparabolic (seaweed) subalgebras of semi-simple Lie algebras. We also provide a survey of Lie algebras for which Dixmier’s problem is known to have a positive answer. This includes all Lie algebras of dimension at most 8. We prove this is also true for all 9-dimensional algebraic Lie algebras. Finally, we improve the statement of Theorem 53 of Ooms (J. Algebra 477, 95–146, 2017).