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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.

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