On the structure of cofree Hopf algebras

Details On the structure of cofree Hopf algebras On the structure of cofree Hopf algebras

2004-05-17

arxiv.org/abs/math/0405330v4

J. reine angew. Math. 592 (2006), 123--155.

Mathematics

Quantum Algebra

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Scientific paper

We prove an analogue of the Poincare'-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 B-infini-algebra. We construct a universal enveloping functor U2 from B-infini-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 simple description of the free 2as-algebra in terms of planar trees to unravel the structure of the operad B-infini.

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