# Generalized Bialgebras and Triples of Operads

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*Jean-Louis Loday*

A publication of the Société Mathématique de France

This book introduces the notion of generalized bialgebra,
which includes the classical notion of bialgebra (Hopf algebra) and
many others, among them the tensor algebra equipped with the
deconcatenation as coproduct. The author proves 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 cocommutative
bialgebras, to a large class of generalized bialgebras.

Technically, the author works in the theory of operads which allows
him to state his main theorem and permits him to give it a conceptual
proof. A generalized bialgebra type is determined by two operads: one
for the coalgebra structure \(\mathcal{C}\) and one for the algebra
structure \(\mathcal{A}\). 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
\(\mathcal{A}\), denoted \(\mathcal{P}\) . The structure
theorem gives conditions under which a connected generalized bialgebra
is cofree (as a connected \(\mathcal{C}\)-coalgebra) and can be
reconstructed out of its primitive part by means of an enveloping
functor from \(\mathcal{P}\)-algebras to
\(\mathcal{A}\)-algebras. The classical case is \((\mathcal
{C, A, P})=(Com, As, Lie)\).

This structure theorem unifies several results, generalizing the PBW and the
CMM theorems, scattered in the literature. The author treats many explicit
examples and suggests a few conjectures.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

#### Readership

Graduate students and research mathematicians interested in algebra.