  To any field K of characteristic 0, we associate a set Sha(K). Elements of
Sha(K) are equivalence classes of families of Lie polynomials subject to
associativity relations. We construct an injection and a retraction between
Sha(K) and the set of quantization functors of Lie bialgebras over K. This
construction involves the following steps. 1) To each element \varpi of Sha(K),
we associate a functor g\mapsto Sh(g) from the category of Lie algebras to that
of Hopf algebras; Sh(g) contains Ug. 2) When g and h are Lie algebras, and
r_{gh} \in g\otimes h, we construct an element R(r_{gh}) of Sh(g)\otimes Sh(h)
satisfying quasitriangularity identities; R(r_{gh}) defines a Hopf algebra
morphism from Sh(g)^* to Sh(h). 3) When g = h and r\in g\otimes g is a solution
of CYBE, we construct a series \rho(r) such that R(\rho(r)) is a solution of
QYBE. The expression of \rho(r) in terms of r 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 g as the
image of the morphism defined by R(\rho(r)), where r\in g\otimes g^* is the
canonical element attached to g.
