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This book is dedicated to the structure and combinatorics of classical Hopf algebras. Its main focus is on commutative and cocommutative Hopf algebras, such as algebras of representative functions on groups and enveloping algebras of Lie algebras, as explored in the works of Borel, Cartier, Hopf and others in the 1940s and 50s.The modern and systematic treatment uses the approach of natural operations, illuminating the structure of Hopf algebras by means of their endomorphisms and their combinatorics. Emphasizing notions such as pseudo-coproducts, characteristic endomorphisms, descent algebras and Lie idempotents, the text also covers the important case of enveloping algebras of pre-Lie algebras. A wide range of applications are surveyed, highlighting the main ideas and fundamental results.
Suitable as a textbook for masters or doctoral level programs, this book will be of interest to algebraists and anyone working in one of the fields of application of Hopf algebras.
List of contents
1. Introduction.- Part I General Theory.- 2 Coalgebras, Duality.- 3. Hopf Algebras and Groups.- 4. Structure Theorems.- 5. Graded Hopf Algebras and the Descent Gebra.- 6. PreLie Algebras.- Part II Applications.- 7. Group Theory.- 8. Algebraic Topology.- 9. Combinatorial Hopf Algebras.- 10. Renormalization.
About the author
Pierre Cartier is a member of the IHES, alumnus of the Ecole Normale Supérieure and former research director at CNRS. A long associate of the Bourbaki group, he is known for his wide range of interests and contributions, among others in algebraic geometry, representation theory and mathematical physics. He is one of the founders of the theory of coalgebras and Hopf algebras.
Frédéric Patras, alumnus of the Ecole Normale Supérieure and research director at CNRS, is an expert in the theory of Hopf algebras and their applications in analysis, combinatorics, Lie theory, probability, theoretical chemistry and physics. He has published and edited over a hundred works on various subjects.
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"The book should be useful for those interesting in to learn about the subject, and also for expert looking for precise reference, new perspectives and applications. ... At the end of each chapter, the reader will find some useful remarks on the bibliography." (Cristian Vay, zbMATH 1514.16001, 2023)
"Each chapter contains a final section with bibliographical indications; these sections contain a lot of history and philosophy of the subject, making the book very attractive to mathematicians who already have a Hopf algebra background. ... the book is written in a self-contained way, making it suitable as a textbook for seminars and even master courses. ... a welcome addition to the bookshelf of any Hopf algebraist." (Stefaan Caenepeel, Mathematical Reviews, January, 2023)
