Classical Hopf Algebras and Their Applications

This book explores the structure and combinatorics of classical Hopf algebras, focusing on commutative and cocommutative Hopf algebras, such as those of representative functions and Lie algebras. It uses the approach of natural operations and covers applications in algebra, geometry, and mathematical physics.

Format: Paperback / softback
Length: 268 pages
Publication date: 22 September 2022
Publisher: Springer Nature Switzerland AG

This book is a comprehensive exploration of the structure and combinatorics of classical Hopf algebras, with a primary focus on commutative and cocommutative Hopf algebras. These algebras are studied in the context of representative functions on groups and enveloping algebras of Lie algebras, as developed by renowned mathematicians such as Borel, Cartier, Hopf, and others during the 1940s and 50s. The modern and systematic approach employed in this text utilizes the concept of natural operations, shedding light on the structure of Hopf algebras through the study of their endomorphisms and combinatorics. Key notions such as pseudo-coproducts, characteristic endomorphisms, descent algebras, and Lie idempotents are introduced and discussed in detail. Moreover, the text covers the important case of enveloping algebras of pre-Lie algebras, providing a comprehensive treatment of this topic.

The book is designed to serve as a textbook for advanced courses in algebra and mathematics, particularly for students pursuing a master's or doctoral degree. It offers a thorough and detailed account of the subject matter, making it accessible to both experts and beginners in the field. The wide range of applications surveyed in the book highlights the fundamental ideas and results obtained in Hopf algebra research. These applications span various areas, including mathematics, physics, computer science, and engineering.

One of the notable features of this book is its emphasis on the geometric aspects of Hopf algebras. The authors use diagrams and visualizations to illustrate the key concepts and relationships, making the material more accessible and engaging for students. Additionally, the book includes a comprehensive bibliography that provides a valuable reference for further reading on the topic.

In conclusion, this book is an essential resource for anyone interested in the structure and combinatorics of classical Hopf algebras. Its comprehensive coverage, systematic approach, and emphasis on geometric aspects make it suitable for both academic and professional purposes. Whether you are a student, researcher, or practitioner in the field, this book will provide you with a deep understanding of Hopf algebras and their applications.

Weight: 438g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030778477
Edition number: 1st ed. 2021