Reflection Groups and Rigidity of Quadratic Poisson Algebras

TL;DR

This paper investigates the invariant theory of quadratic Poisson algebras, establishing a Poisson analogue of the Shephard-Todd-Chevalley theorem and revealing rigidity phenomena related to group actions and fixed subalgebras.

Contribution

It proves a Poisson version of the Shephard-Todd-Chevalley theorem for skew-symmetric brackets and demonstrates rigidity results for various quadratic Poisson algebras.

Findings

Fixed Poisson subring is skew-symmetric iff the group is generated by reflections.

Many quadratic Poisson algebras have limited or no reflections in their automorphism groups.

Fixed Poisson subring is generally not isomorphic to the original algebra unless the group is trivial.

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.