On Dixmier's fourth problem

TL;DR

This paper investigates Dixmier's fourth problem, showing that for certain algebraic Lie algebras with polynomial Poisson semi-centers, the center of their division ring is purely transcendental, and surveys known positive cases.

Contribution

It proves the conjecture for algebraic Lie algebras with polynomial Poisson semi-centers and extends known results to 9-dimensional algebraic Lie algebras.

Findings

Center Z(D(L)) is purely transcendental for algebraic L with polynomial Sy(L)

All Lie algebras of dimension ≤8 satisfy Dixmier's problem positively

All 9-dimensional algebraic Lie algebras satisfy Dixmier's problem

Abstract

Let L be a finite dimensional Lie algebra over an algebraically closed field k of characteristic zero. Denote by U(L) its enveloping algebra with quotient division ring D(L). 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(L)) is always a purely transcendental extension of k. We show this is the case if L is algebraic whose Poisson semi-center Sy(L) is a polynomial algebra over k. This can be applied to many (bi)parabolic 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 Theorem 53 of [45].