Beyond Hopf algebras

In the theory of Hopf algebras there is a structure theorem which is very useful. It says that, in characteristic zero, a connected cocommutative Hopf algebra is cofree as a coalgebra and is, as an algebra, isomorphic to the universal enveloping algebra of a Lie algebra. This structure theorem is essentially equivalent to the union of the Poincaré-Birkhoff-Witt theorem with the Cartier-Milnor- Moore theorem. It involves three types of algebras, that is three operads: Com for the coalgebra structure, As for the algebra structure, and Lie for the structure of the primitive part. The purpose of this series of talks is to show that there are numerous other examples of this form, many of them already in the literature. We give elementary conditions on a triple of operads (C, A, P) so that there is a structure theorem for C^c-A-bialgebras, the primitive part being a P-algebra. Then it is called a good triple. The paradigm is (Com, As, Lie).

In many cases the C^c-A-bialgebras are, in fact, combinatorial Hopf algebras (with more structure).