Generalized bialgebras and triples of operads

By Jean-Louis Loday. Astérisque, 2006

103 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e3aca077b5d470624ea6567cdc26530251023d7a

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In Algebra and Number Theory

1. Wolfgang Alexander Moens
2. Dietrich Burde
3. Manuel Ceballos
4. Alice Fialowski

In Geometry and Topology

1. Wolfgang Alexander Moens
2. Dietrich Burde
3. Manuel Ceballos
4. Alice Fialowski

In Mathematical Physics

1. Wolfgang Alexander Moens
2. Dietrich Burde
3. Manuel Ceballos
4. Alice Fialowski

Citations in the corpus

• Commutative post-Lie algebra structures on Lie algebras (2015, background): “Loday [14] studied pre-Lie algebras and post-Lie algebras within the context of algebraic operad triples.”