On the invertibility of quantization functors

Benjamin Enriquez (1) , Pavel Etingof

1 IRMA - Institut de Recherche Mathématique Avancée

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 derive from there a classification of all twistors killing a given associator. 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.

Domaines: Algèbres quantiques [math.QA]

Benjamin Enriquez, Pavel Etingof. On the invertibility of quantization functors. Journal of Algebra, 2005, 289, num. 2, pp.321-345. ⟨10.1016/j.jalgebra.2005.01.056⟩. ⟨hal-00087043⟩