On the Quantization-Dequantization Correspondence for (co)Poisson Hopf Algebras
Andrea Rivezzi, Jonas Schnitzer
TL;DR
The paper develops a broad, category-theoretic framework for quantization-dequantization of (co)Poisson Hopf algebras by constructing functorial quantizations and dequantizations via Drinfeld associators and the Grothendieck-Teichmüller semigroup. It introduces Drinfeld-Yetter categories over (co)Poisson Hopf monoids, along with adapted/coadapted functors that generate Hopf structures and their module categories, and proves a functorial equivalence between quantized and dequantized worlds. The framework recovers Etingof-Kazhdan results as special cases and extends deformation quantization to Tamarkin’s approach to Deligne’s conjecture, providing explicit constructions that operate in a flexible, fully categorical setting. The methodology broadens the landscape for deformation theory, enabling quantization of Poisson Hopf algebras and their modules and suggesting further applications in Lie bialgebras, Lie algebroids, and related geometric representation theory.$
Abstract
In this paper, we construct a functorial quantization of (co)Poisson Hopf algebras within a broad categorical framework. We further introduce categories naturally associated with (co)Poisson Hopf algebras, namely Drinfeld-Yetter modules. These categories provide a canonical setting in which we define explicit dequantization functors that are inverse to the quantization functors. Using this framework, we also establish functorial (de)quantization results for the corresponding module categories. Finally, we recover the classical results of Etingof and Kazhdan as special cases of our construction and discuss applications to deformation quantization à la Tamarkin.