**Astérisque**

Volume: 320;
2008;
114 pp;
Softcover

MSC: Primary 16; 17; 18; 81;
**Print ISBN: 978-2-85629-257-0
Product Code: AST/320**

List Price: $42.00

Individual Member Price: $37.80

# 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.