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Generalized Bialgebras and Triples of Operads

出版	Société mathématique de France, 2008
主題	Mathematics / Algebra / GeneralMathematics / Algebra / AbstractMathematics / Group Theory
ISBN	28562925779782856292570
URL	http://books.google.com.hk/books?id=J44_AQAAIAAJ&hl=&source=gbs_api

註釋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 Poincare-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.