Lecture Notes in Mathematics

Hopf Algebras and Their Generalizations from a Category Theoretical Point of View

Authors: Böhm, Gabriella

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  • First book to provide a unified category theoretical description of apparently different generalizations of Hopf algebras (including Hopf algebroids, weak Hopf algebras, and Hopf bimonoids in duoidal categories)
  • Includes fully worked solutions to the exercises
  • Enjoys a rigorous Bourbaki-style presentation, supplemented with various examples
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eBook £27.99
price for United Kingdom (gross)
  • ISBN 978-3-319-98137-6
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Softcover £34.99
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  • ISBN 978-3-319-98136-9
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  • Usually dispatched within 3 to 5 business days.
About this book

These lecture notes provide a self-contained introduction to a wide range of generalizations of Hopf algebras. Multiplication of their modules is described by replacing the category of vector spaces with more general monoidal categories, thereby extending the range of applications.

Since Sweedler's work in the 1960s, Hopf algebras have earned a noble place in the garden of mathematical structures. Their use is well accepted in fundamental areas such as algebraic geometry, representation theory, algebraic topology, and combinatorics. Now, similar to having moved from groups to groupoids, it is becoming clear that generalizations of Hopf algebras must also be considered. This book offers a unified description of Hopf algebras and their generalizations from a category theoretical point of view. The author applies the theory of liftings to Eilenberg–Moore categories to translate the axioms of each considered variant of a bialgebra (or Hopf algebra) to a bimonad (or Hopf monad) structure on a suitable functor. Covered structures include bialgebroids over arbitrary algebras, in particular weak bialgebras, and bimonoids in duoidal categories, such as bialgebras over commutative rings, semi-Hopf group algebras, small categories, and categories enriched in coalgebras.

Graduate students and researchers in algebra and category theory will find this book particularly useful. Including a wide range of illustrative examples, numerous exercises, and completely worked solutions, it is suitable for self-study.

About the authors

Gabriella Böhm received her master's degree in 1993, and a PhD in 1999 from the Eötvös University in Budapest. She has a broad expertise in generalizations of Hopf algebra, and has made significant contributions to the theory of Hopf algebroids as one of the inventors of weak Hopf algebra. The key feature of her work is the use of category theoretical methods in treating algebraic questions.

Reviews

“The main achievements of the book is to derive the axioms defining a given Hopf algebraic structure from the mentioned feature of its category of modules. ... The book offers a self-contained presentation, starting from the basic notions of categories and functors. The introduction of the different algebraic structures is illustrated with several examples and there is an extensive list of bibliographical references.” (Sonia Natale, zbMath 1417.16034, 2019)


Table of contents (9 chapters)

Table of contents (9 chapters)

Buy this book

eBook £27.99
price for United Kingdom (gross)
  • ISBN 978-3-319-98137-6
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover £34.99
price for United Kingdom (gross)
  • ISBN 978-3-319-98136-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Hopf Algebras and Their Generalizations from a Category Theoretical Point of View
Authors
Series Title
Lecture Notes in Mathematics
Series Volume
2226
Copyright
2018
Publisher
Springer International Publishing
Copyright Holder
Springer Nature Switzerland AG
eBook ISBN
978-3-319-98137-6
DOI
10.1007/978-3-319-98137-6
Softcover ISBN
978-3-319-98136-9
Series ISSN
0075-8434
Edition Number
1
Number of Pages
XI, 165
Number of Illustrations
239 b/w illustrations
Topics