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On the structure of cofree Hopf algebras. (English) Zbl 1096.16019
The Cartier-Milnor-Moore theorem says that a connected cocommutative Hopf algebra \(H\) in characteristic 0 is isomorphic to \(U(\text{Prim\,}H)\), where \(\text{Prim\,}H\) is the Lie algebra of primitive elements, and \(U\) is the universal enveloping functor. The authors give an analogue of this result and of the Poincaré-Birkhoff-Witt theorem in the non-cocommutative case. If \(H\) is a bialgebra over an arbitrary field, \(\text{Prim\,}H\) is regarded as a non-differential \(B_\infty\)-algebra. The concepts of 2-associative algebra and 2-associative bialgebra are introduced, and a universal enveloping functor \(U2\) is constructed from non-differential \(B_\infty\)-algebras to 2-associative algebras.
The main result says that for a bialgebra \(H\) the following are equivalent: (a) \(H\) is a connected 2-associative bialgebra; (b) \(H\) is isomorphic to \(U2(\text{Prim\,}H)\) as 2-associative bialgebras; (c) \(H\) is cofree among the connected coalgebras. As a consequence, it is obtained that a cofree Hopf algebra is of the form \(U2(R)\) for a non-differential \(B_\infty\)-algebra \(R\). A key result in the proof is that a connected unital infinitesimal bialgebra is isomorphic to the tensor algebra equipped with the deconcatenation product. Finally, an explicit description of the free 2-associative algebra is given in terms of planar trees, and the operad associated to 2-associative algebras is studied.

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16S30 Universal enveloping algebras of Lie algebras
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