Invariants of Quantizations of Unimodular Quadratic Polynomial Poisson Algebras of Dimension 3

Abstract

Let $P=\mathbb{k}[x_1,x_2,x_3]$ be a unimodular quadratic Poisson algebra, with its Poisson bracket written as $\{x_i,x_j\}=\sum _{k,l}{c}_{i,j}^{k,l}{x}_k{x}_l$, $1\le i<j\le 3$. Let $P_{\hslash }$ be the deformation quantization of $P$ constructed as follows: $P_{\hslash }=\mathbb{k}⟨y_1,y_2,y_3⟩/([y_i,y_j]=\frac{\hslash }{2}\sum _{k,l}{c}_{i,j}^{k,l}(y_k{y}_l+y_l{y}_k){)}_{1\le i<j\le 3}$. In this paper, we establish that $P$ and $P_{\hslash }$ possess identical graded automorphisms and reflections, and that taking invariant subalgebras and taking deformation quantizations are two commutative processes.