Serie : Astérisque. Vol 320
Paru le 15/04/2009 | Broché IX-116 pages
Professionnels
We introduce the notion of generalized bialgebra, which includes the classical notion of bialgebra (Hopf algebra) and many others, like, for instance, the tensor algebra equipped with the deconcatenation as coproduct. We prove that, under some mild conditions, a connected generalized bialgebra is completely determined by its primitive part. This structure theorem extends the classical Poincaré-Birkhoff-Witt theorem and Cartier-Milnor-Moore theorem, valid for co-commutative bialgebras, to a large class of generalized bialgebras. Technically we work in the theory of operads which allows us to state our main results and permits us to give it a conceptual proof. A generalized bialgebra type is determined by two operads: one for the coalgebra structure C, and one for the algebra structure A. There is also a compatibility relation relating the two. Under some conditions, the primitive part of such a generalized bialgebra is an algebra over some sub-operad of A, denoted P. The structure theorem gives conditions under which a connected generalized bialgebra is cofree (as a connected C-coalgebra) and can be re-constructed out of its primitive part by means of an enveloping functor from P-algebras to A-algebras. The classical case is (C, A, P) = (Com, As, Lie). This structure theorem unifies several results, generalizing the PBW and the CMM theorems, scattered in the literature. We treat many explicit examples and suggest a few conjectures.
