Quantization of Lie bialgebras and shuffle algebras of Lie algebras

To any field Bbb K Bbb K of characteristic zero, we associate a set (mathbbK) (mathbb{K}) and a group G₀(Bbb K) {cal G}_0(Bbb K) . Elements of (mathbbK) (mathbb{K}) are equivalence classes of families of Lie polynomials subject to associativity relations. Elements of G₀(Bbb K) {cal G}_0(Bbb K) are universal automorphisms of the adjoint representations of Lie bialgebras over Bbb K Bbb K . We construct a bijection between (mathbbK)×G₀(Bbb K) (mathbb{K})times{cal G}_0(Bbb K) and the set of quantization functors of Lie bialgebras over Bbb K Bbb K . This construction involves the following steps.? 1) To each element v varpi of (mathbbK) (mathbb{K}) , we associate a functor frak a?operatornameSh^v(frak a) frak amapstooperatorname{Sh}^varpi(frak a) from the category of Lie algebras to that of Hopf algebras; operatornameSh^v(frak a) operatorname{Sh}^varpi(frak a) contains Ufrak a Ufrak a .? 2) When frak a frak a and frak b frak b are Lie algebras, and r_{frak afrak b} ? frak a?frak b r_{frak afrak b} infrak aotimesfrak b , we construct an element ?^v (r_{frak afrak b}) {cal R}^{varpi} (r_{frak afrak b}) of operatornameSh^v(frak a)?operatornameSh^v(frak b) operatorname{Sh}^varpi(frak a)otimesoperatorname{Sh}^varpi(frak b) satisfying quasitriangularity identities; in particular, ?^v(r_{frak afrak b}) {cal R}^varpi(r_{frak afrak b}) defines a Hopf algebra morphism from operatornameSh^v(frak a)^* operatorname{Sh}^varpi(frak a)^* to operatornameSh^v(frak b) operatorname{Sh}^varpi(frak b) .? 3) When frak a = frak b frak a = frak b and r_frak a ? frak a?frak a r_frak ainfrak aotimesfrak a is a solution of CYBE, we construct a series r^v(r_frak a) rho^varpi(r_frak a) such that ?^v(r^v(r_frak a)) {cal R}^varpi(rho^varpi(r_frak a)) is a solution of QYBE. The expression of r^v(r_frak a) rho^varpi(r_frak a) in terms of r_frak a r_frak a involves Lie polynomials, and we show that this expression is unique at a universal level. This step relies on vanishing statements for cohomologies arising from universal algebras for the solutions of CYBE.? 4) We define the quantization of a Lie bialgebra frak g frak g as the image of the morphism defined by ?^v(r^v(r)) {cal R}^varpi(rho^varpi(r)) , where r ? mathfrakg ?mathfrakg^* r in mathfrak{g} otimes mathfrak{g}^* .