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On Dixmier’s Fourth Problem

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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 “Algèbres 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).

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Acknowledgments

We would like to thank Michel Van den Bergh for some helpful discussions and for providing Proposition 3.1. We are also grateful to Vladimir Popov for accurately describing the relevant results (and their references) of Katsylo and Bogomolov.

Finally we wish to thank Viviane Mebis for the excellent typing of the manuscript.

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  1. Mathematics Department, Hasselt University, Agoralaan, Campus Diepenbeek, 3590, Diepenbeek, Belgium

    Alfons I. Ooms

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  1. Alfons I. Ooms
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Correspondence to Alfons I. Ooms.

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Presented by: Michel Van den Bergh

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Ooms, A.I. On Dixmier’s Fourth Problem. Algebr Represent Theor 25, 561–579 (2022). https://doi.org/10.1007/s10468-021-10035-z

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