Quantization of Lie bialgebras and shuffle algebras of Lie algebras

Abstract

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.