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On the Semi-centre of a Poisson Algebra

Abstract

If \(\mathfrak {g}\) is a Lie algebra then the semi-centre of the Poisson algebra \(S(\mathfrak {g})\) is the subalgebra generated by \(\operatorname {ad}(\mathfrak {g})\)-eigenvectors. In this paper we abstract this definition to the context of integral Poisson algebras. We identify necessary and sufficient conditions for the Poisson semi-centre Asc to be a Poisson algebra graded by its weight spaces. In that situation we show the Poisson semi-centre exhibits many nice properties: the rational Casimirs are quotients of Poisson normal elements and the Poisson Dixmier–Mœglin equivalence holds for Asc.

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Acknowledgements

We would like to thank Professor David Jordan for carefully reading the first draft of this manuscript and making helpful suggestions. The second author is grateful for the support of EPSRC grant EP/N034449/1.

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Correspondence to Cesar Lecoutre.

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Presented by: Michel Van den Bergh

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Lecoutre, C., Topley, L. On the Semi-centre of a Poisson Algebra. Algebr Represent Theor 23, 875–886 (2020). https://doi.org/10.1007/s10468-019-09879-3

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Keywords

  • Poisson algebra
  • Semi-invariant theory
  • Poisson Dixmier–Moeglin equivalence