Generalized bialgebras and triples of operads

Details Generalized bialgebras and triples of operads Generalized bialgebras and triples of operads

2006-11-28

arxiv.org/abs/math/0611885v3

Aste'risque 320 (2008), vi+114 pp.

Mathematics

Quantum Algebra

Slight modification of the quotient triple proposition (3.1.1). Typos corrected. 110 pages

Scientific paper

We introduce the notion of generalized bialgebra, which includes the classical notion of bialgebra (Hopf algebra) and many others. We prove that, under some mild conditions, a connected generalized bialgebra is completely determined by its primitive part. This structure theorem extends the classical Poincar\'e-Birkhoff-Witt theorem and the Cartier-Milnor-Moore theorem, valid for cocommutative bialgebras, to a large class of generalized bialgebras. Technically we work in the theory of operads which permits us to give a conceptual proof of our main theorem. It unifies several results, generalizing PBW and CMM, scattered in the literature. We treat many explicit examples and suggest a few conjectures.

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