Tressages des groupe de Poisson formels à dual quasitriangulaire

Abstract

Let $\mathfrak{g}$ be a quasitriangular Lie bialgebra over a field $K$ of characteristic zero, and let ${\mathfrak{g}}^{\ast }$ be its dual Lie bialgebra. We prove that the formal Poisson group $K\left[\right[{\mathfrak{g}}^{\ast }\left]\right]$ is a braided Hopf algebra, thus generalizing a result due to Reshetikhin (in the case $\mathfrak{g}=\mathfrak{sl}(2,K)$). The proof is via quantum groups, using the existence of a quasitriangular quantization of ${\mathfrak{g}}^{\ast }$, as well as the fact that this one provides also a quantization of $K\left[\right[{\mathfrak{g}}^{\ast }\left]\right]$.

AMS-TeX file, 13 pages, in French; this is the authors' file of the final version (after the refereeing process), as sent for publication. There exists also an English version, posted on arXiv as preprint math/9909065

References in corpus (2)

• Quantization of Lie bialgebras, V
• The $R$--matrix action of untwisted affine quantum groups at roots of 1

Cited by in corpus (4)

• Braiding structures on formal Poisson groups and classical solutions of the QYBE
• On the unicity of braidings of quasitriangular Lie bialgebras
• Poisson algebras associated to quasi-Hopf algebras
• A h-adic valuation property of universal R-matrices