Invariants of Poisson Algebras, Poisson Enveloping Algebras, and Deformation Quantizations

Abstract

The Shephard-Todd-Chevalley Theorem and the Watanabe Theorem are among the earliest results addressing the homological properties of invariant subalgebras. Initially studied in the context of polynomial algebras, these theorems have motivated researchers to generalize their applicability beyond the scope of commutative algebras. Notable instances include, but certainly are not limited to: Alev and Polo's studies on enveloping algebra of semisimple Lie algebras and Weyl algebras; Kirkman, Kuzmanovich, and Zhang's studies on skew polynomial rings, quantum matrix algebras, non-PI Sklyanin algebras and down up algebras; Gaddis, Veerapen, and Wang's studies on semiclassical limits (Poisson algebras) of several families of Artin-Schelter regular algebras. In this dissertation, we will continue Gaddis, Veerapen, and Wang's studies on Poisson algebras, a commutative algebra together with a non-commutative bracket. Our primary emphasis will be on quadratic Poisson structures on polynomial rings of three variables. Our objective is to prove variants of the Shephard-Todd-Chevalley Theorem for these Poisson algebras and their associated algebraic structures: Poisson enveloping algebras and deformation quantizations. Furthermore, we will prove a variant of the Watanabe Theorem for Poisson enveloping algebras arising from quadratic Poisson structures on an arbitrary polynomial ring.