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: Hardback
Length: 268 pages
Publication date: 21 September 2021
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 1950s. 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 master's or doctoral degrees. It offers a thorough and detailed account of the subject matter, making it valuable to algebraists and researchers working in various fields where Hopf algebras find applications.

Throughout the book, numerous examples and applications are surveyed, highlighting the main ideas and fundamental results. These applications range from algebraic geometry and mathematical physics to computer science and theoretical biology. By presenting these applications, the book demonstrates the wide-ranging significance and practical applications of Hopf algebras in modern mathematics.

In conclusion, this book is an essential resource for anyone interested in the structure and combinatorics of classical Hopf algebras. It provides a modern and comprehensive treatment of the subject matter, making it suitable for both academic and professional purposes.

Weight: 594g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030778446
Edition number: 1st ed. 2021