# Reflection Groups and Rigidity of Quadratic Poisson Algebras

```@article{Gaddis2021ReflectionGA,
title={Reflection Groups and Rigidity of Quadratic Poisson Algebras},
journal={Algebras and Representation Theory},
year={2021}
}```
• Published 16 June 2020
• Mathematics
• Algebras and Representation Theory
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…
3 Citations

### Twists of graded Poisson algebras and related properties

• Mathematics
• 2022
. We introduce a Poisson version of the graded twist of a graded as- sociative algebra and prove that every graded Poisson structure on a connected graded polynomial ring A := k [ x 1 ,...,x n ] is a

### Cancellation and skew cancellation for Poisson algebras

• Mathematics
Mathematische Zeitschrift
• 2022
. We study the Zariski cancellation problem for Poisson algebras in three variables. In particular, we prove those with Poisson bracket either being quadratic or derived from a Lie algebra are

### A ug 2 02 1 CANCELLATION AND SKEW CANCELLATION FOR POISSON ALGEBRAS

• Mathematics
• 2021
We study the Zariski cancellation problem for Poisson algebras in three variables. In particular, we prove those with Poisson bracket either being quadratic or derived from a Lie algebra are

## References

SHOWING 1-10 OF 40 REFERENCES

### Finite Group Actions on Poisson Algebras

• Mathematics
• 2003
Let Andenote the Weyl algebra of all differential operators on the polynomial algebra C[X1,… Xn].It is well known that if G is a finite group of algebra automorphisms of An, then An is a simple

### Derived invariants of the fixed ring of enveloping algebras of semisimple Lie algebras

Let \$\${\mathfrak {g}}\$\$ g be a semisimple complex Lie algebra, and let W be a finite subgroup of \$\${\mathbb {C}}\$\$ C -algebra automorphisms of the enveloping algebra \$\$U({\mathfrak {g}})\$\$ U ( g ) .

### A Rigidity Theorem for Finite Group Actions on Enveloping Algebras of Semisimple Lie Algebras

• Mathematics
• 1995
Abstract Let g denote a semisimple Lie algebra over an algebraically closed field k of characteristic zero and G , a finite group of k -automorphisms of the enveloping algebra U of g . In this paper,

### Poisson Polynomial Rings

Let A be a Poisson algebra with Poisson bracket {·, ·} A and let α,δ be linear maps from A into itself. Here we find a necessary and sufficient condition for the pair (α,δ) such that the polynomial

### On one class of exact Poisson structures

We discuss some properties of a natural class of Poisson structures on Euclidean spaces and abstract manifolds. In particular it is proved that such structures are always exact and may be

### Polynomial Invariants of Finite Groups

Written by an algebraic topologist motivated by his own desire to learn, this well-written book represents the compilation of the most essential and interesting results and methods in the theory of

### Polynomial invariants of finite groups

1. Finite generation of invariants 2. Poincare series 3. Divisor classes, ramification and hyperplanes 4. Homological properties of invariants 5. Polynomial tensor exterior algebra 6. Polynomial