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arXiv · 2006.09280

Reflection Groups and Rigidity of Quadratic Poisson Algebras

Jason Gaddis·Padmini Veerapen·Xingting Wang
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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 A^G 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 A^G is not isomorphic to A as Poisson algebras unless G is trivial.

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Jason Gaddis, Padmini Veerapen, Xingting Wang. 2020-06-16. Reflection Groups and Rigidity of Quadratic Poisson Algebras. https://doi.org/10.1007/s10468-021-10096-0

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