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

2023/11/29 by Chengyuan Ma, Ma, Chengyuan
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2311.17385

openalex publication_date 2023/11/29 · openalex created_date 2023/12/01 · openalex updated_date 2026/07/28

Abstract

Let P = \Bbbk[x₁, x₂, x₃] be a unimodular quadratic Poisson algebra, with its Poisson bracket written as \x_i, x_j\ = ∑_k,lc_i,j^k,lx_kx_l, 1 ≤ i < j ≤ 3. Let P_ℏ be the deformation quantization of P constructed as follows: P_ℏ = \Bbbk⟨ y₁, y₂, y₃⟩/([y_i,y_j]=(ℏ)/(2)∑_k,lc_i,j^k,l(y_ky_l+y_ly_k))_{1 ≤ i < j ≤ 3}. In this paper, we establish that P and P_ℏ possess identical graded automorphisms and reflections, and that taking invariant subalgebras and taking deformation quantizations are two commutative processes.

Related

• Quantization of quadratic Poisson brackets on a polynomial algebra of three variables
• Deformation Quantization of Polynomial Poisson Algebras
• Invariants of Unimodular Quadratic Polynomial Poisson Algebras of Dimension 3
• Deformation quantization of integrable systems
• Identities for deformation quantizations of almost Poisson algebras