Categorical Perspective on Quantization of Poisson Algebra

Abstract

We propose a generalization of quantization as a categorical way. For a fixed Poisson algebra quantization categories are defined as subcategories of R-module category with the structure of classical limits. We construct the generalized quantization categories including matrix regularization, strict deformation quantization, prequantization, and Poisson enveloping algebra, respectively. It is shown that the categories of strict deformation quantization, prequantization, and matrix regularization with some conditions are categorical equivalence. On the other hand, the categories of Poisson enveloping algebra is not equivalent to the other categories.

20 pages; In v3, some inaccuracies are corrected and preconditions in propositions 2.5, 7.1, 7.2, 7.3, and 7.7 are added. Definitions and theorems in section 5 are revised. Theorem 4.6 is added. Appendix A and B are added. v3 has minor changes in definition 3.1 and 4.1