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

Abstract

Let $P = \Bbbk[x_1, x_2, x_3]$ be a unimodular quadratic Poisson algebra, with its Poisson bracket written as $\{x_i, x_j\} = \displaystyle{\sum_{k,l}c_{i,j}^{k,l}x_kx_l}$, $1 \leq i < j \leq 3$. Let $P_{\hbar}$ be the deformation quantization of $P$ constructed as follows: $P_{\hbar} = \Bbbk\langle y_1, y_2, y_3\rangle/([y_i,y_j]=\frac{\hbar}{2}\displaystyle{\sum_{k,l}}c_{i,j}^{k,l}(y_ky_l+y_ly_k))_{1 \leq i < j \leq 3}$. In this paper, we establish that $P$ and $P_{\hbar}$ possess identical graded automorphisms and reflections, and that taking invariant subalgebras and taking deformation quantizations are two commutative processes.