Braidings of Poisson groups with quasitriangular dual (Tressages des groupes de Poisson à dual quasitriangulaire)

Details Braidings of Poisson groups with quasitriangular dual (Tressages des groupes de Poisson à dual quasitriangulaire) Braidings of Poisson groups with quasitriangular dual (Tressages des groupes de Poisson à dual quasitriangulaire)

1999-09-13

arxiv.org/abs/math/9909065v2

"Tressages des groupes de Poisson formels a` dual quasitriangulaire", Journal of Pure and Applied Algebra 161 (2001), 295-307

Mathematics

Quantum Algebra

11 pages, AMS-TeX file; English and French version available. The whole paper (even the title) has been entirely rewritten. So

Scientific paper

Let g be a quasitriangular Lie bialgebra over a field k of characteristic zero, and let g^* be its dual Lie bialgebra. We prove that the formal Poisson group F[[g^*]] is a braided Hopf algebra. More generally, we prove that if (U_h,R) is any quasitriangular QUEA, then (U_h', Ad(R)|_{U_h' \otimes U_h'}) --- where U_h' is defined by Drinfeld --- is a braided QFSHA. The first result is then just a consequence of the existence of a quasitriangular quantization (U_h,R) of U(g) and of the fact that U_h' is a quantization of F[[g^*]]. ----- Soit g une big\`ebre de Lie quasitriangulaire sur un corps k de characteristique zero, et soit g^* sa big\`ebre de Lie duale. Nous prouvons que le groupe de Poisson formel F[[g^*]] est une algebre de Hopf tress\'ee. Plus en g\'en\'eral, nous prouvons que, si (U_h,R) est une QUEA quasitriangulaire, alors (U_h', Ad(R)|_{U_h' \otimes U_h'}) --- o\`u U_h' est definie par Drinfeld --- est une QFSHA tress\'ee. Le premier r\'esultat est alors une consequence de l'existence d'une quantification quasitriangulaire (U_h,R) de U(g) et du fait que U_h' est une quantification de F[[g^*]].

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