On the invertibility of quantization functors

TL;DR

This paper proves that certain quantization functors are invertible by using morphisms between props and Hensel's lemma, simplifying proofs of equivalences in quantum algebra and constructing dequantizations.

Contribution

It introduces a new, simplified proof of the invertibility of Etingof-Kazhdan quantization functors using prop morphisms and Hensel's lemma, and extends the approach to dequantizations.

Findings

Etingof-Kazhdan quantization functors are equivalences of categories.

Constructed dequantizations of solutions to the quantum Yang-Baxter equation.

Proved that the prop of QUE algebras is a flat deformation of co-Poisson universal enveloping algebras.

Abstract

Certain quantization problems are equivalent to the construction of morphisms from "quantum" to "classical" props. Once such a morphism is constructed, Hensel's lemma shows that it is in fact an isomorphism. This gives a new, simple proof that any Etingof-Kazhdan quantization functor is an equivalence of categories between quantized universal enveloping (QUE) algebras and Lie bialgebras over a formal series ring (dequantization). We apply the same argument to construct dequantizations of formal solutions of the quantum Yang-Baxter equation and of quasitriangular QUE algebras. We also give structure results for the props involved in quantization of Lie bialgebras, which yield an associator-independent proof that the prop of QUE algebras is a flat deformation of the prop of co-Poisson universal enveloping algebras.