On quantization functors of Lie bialgebras (Q1862246)

This is a survey on recent work on quantization functors of Lie bialgebras. Let QUE be the category of quantized universal enveloping algebras over a field of characteristic 0, and let LBA be the category of Lie bialgebras. The operation of taking the semiclassical limit defines a functor \(\text{SC}:\text{QUE}\rightarrow \text{LBA}\). A quantization functor is a functor \(Q:\text{LBA}\rightarrow \text{QUE}\) such that \(\text{SC}\circ Q\) is isomorphic to the identity. It is shown how the set of quantization functors has a structure of a torsor, and that the Etingof-Kazhdan map [\textit{P. Etingof} and \textit{D. Kazhdan}, Sel. Math., New Ser. 2, 1--41 (1996; Zbl 0863.17008)] is a morphism of torsors. The infinitesimal of this map is computed, and as a consequence it is shown that the quantization functors of finite-dimensional Lie bialgebras are independent of the choice of an associator.