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Reflection Groups and Rigidity of Quadratic Poisson Algebras

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Abstract

In this paper, we study the invariant theory of quadratic Poisson algebras. Let G be a finite group of the graded Poisson automorphisms of a quadratic Poisson algebra A. When the Poisson bracket of A is skew-symmetric, a Poisson version of the Shephard-Todd-Chevalley theorem is proved stating that the fixed Poisson subring AG is skew-symmetric if and only if G is generated by reflections. For many other well-known families of quadratic Poisson algebras, we show that G contains limited or even no reflections. This kind of Poisson rigidity result ensures that the corresponding fixed Poisson subring AG is not isomorphic to A as Poisson algebras unless G is trivial.

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Acknowledgements

The authors thank Akaki Tikaradze for pointing out his paper [31], which resolved one of the questions in an early draft of this paper, and his willingingess to allow the inclusion of Remark 5.13. The authors also appreciate the referee’s close reading and suggestions for improving the manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Miami University, Oxford, OH, 45056, USA

    Jason Gaddis

  2. Department of Mathematics, Tennessee Technological University, Cookeville, TN, 38505, USA

    Padmini Veerapen

  3. Department of Mathematics, Howard University, Washington, DC, 20059, USA

    Xingting Wang

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  1. Jason Gaddis
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  2. Padmini Veerapen
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  3. Xingting Wang
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Correspondence to Jason Gaddis.

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Presented by: Kenneth Goodearl

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The third author was partially supported by Simons Collaboration Grant No. 688403

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Gaddis, J., Veerapen, P. & Wang, X. Reflection Groups and Rigidity of Quadratic Poisson Algebras. Algebr Represent Theor 26, 329–358 (2023). https://doi.org/10.1007/s10468-021-10096-0

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