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Title: | On Dixmier’s Fourth Problem | Authors: | OOMS, Alfons | Issue Date: | 2022 | Publisher: | SPRINGER | Source: | ALGEBRAS AND REPRESENTATION THEORY, 25(3), p. 561-579 | Abstract: | Let g be a finite dimensional Lie algebra over an algebraically closed field k of characteristic zero. Denote by U(g) its enveloping algebra with quotient division ring D(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(g)) is always a purely transcendental extension of k. We show this is the case if g is algebraic whose Poisson semi-center Sy(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). | Notes: | Ooms, AI (corresponding author), Hasselt Univ, Dept Math, Campus Diepenbeek, B-3590 Diepenbeek, Belgium. alfons.ooms@uhasselt.be |
Other: | Ooms, AI (corresponding author), Hasselt Univ, Dept Math, Campus Diepenbeek, B-3590 Diepenbeek, Belgium. alfons.ooms@uhasselt.be | Keywords: | Dixmier's fourth problem;Poisson semi-center;Biparabolic subalgebras | Document URI: | http://hdl.handle.net/1942/34195 | ISSN: | 1386-923X | e-ISSN: | 1572-9079 | DOI: | 10.1007/s10468-021-10035-z | ISI #: | WOS:000626374300001 | Rights: | The Author(s), under exclusive licence to Springer Nature B.V. part of Springer Nature 2021 | Category: | A1 | Type: | Journal Contribution | Validations: | ecoom 2022 |
Appears in Collections: | Research publications |
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