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Jean-Louis Loday, Mara Ronco

On the structure of cofree Hopf algebras

We prove an analogue of the Poincaré-Birkhoff-Witt theorem and of the Cartier-Milnor-Moore theorem for non-cocommutative Hopf algebras. The primitive part of a cofree Hopf algebra is a nondifferential B_∞-algebra. We construct a universal enveloping functor U2 from nondifferential B_∞-algebras to 2-associative algebras, i.e. algebras equipped with two associative operations. We show that any cofree Hopf algebra ℋ is of the form U2(Prim ℋ). We take advantage of the description of the free 2as-algebra in terms of planar trees to unravel the operad associated to nondifferential B_∞-algebras.

Journal fur die reine und angewandte Mathematik (Crelles Journal), Walter de Gruyter

Print ISSN: 0075-4102
Volume: 2006, 03/2006
Pages: 123 - 155

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