Quantifications des algèbres de Hopf d'arbres plans décorés et lien avec les groupes quantiques

We introduce a functor from the category of braided spaces into the category of braided Hopf algebras which associates to a braided space V a braided Hopf algebra of planar rooted trees . We show that the Nichols algebra of V is a subquotient of . We construct a Hopf pairing between and , generalising one of the results of Bull. Sci. Math. 126 (2002) 193-239]. When the braiding of c is given by c(v_i⊗v_j)=q_i,jv_j⊗v_i, we obtain a quantification of the Hopf algebras introduced in Bull. Sci. Math. 126 (2002) 193-239; 126 (2002) 249-288]. When q_i,j=q^a_i,j, with q an indeterminate and (a_i,j)_i,j the Cartan matrix of a semi-simple Lie algebra , then is a subquotient of . In this case, we construct the crossed product of with a torus and then the Drinfel'd quantum double of this Hopf algebra. We show that is a subquotient of .