Overview
- 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
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2226)
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Table of contents (9 chapters)
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Front Matter
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Back Matter
Keywords
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.
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Bibliographic Information
Book Title: Hopf Algebras and Their Generalizations from a Category Theoretical Point of View
Authors: Gabriella Böhm
Series Title: Lecture Notes in Mathematics
DOI: https://doi.org/10.1007/978-3-319-98137-6
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2018
Softcover ISBN: 978-3-319-98136-9Published: 02 November 2018
eBook ISBN: 978-3-319-98137-6Published: 01 November 2018
Series ISSN: 0075-8434
Series E-ISSN: 1617-9692
Edition Number: 1
Number of Pages: XI, 165
Number of Illustrations: 239 b/w illustrations
Topics: Category Theory, Homological Algebra, Associative Rings and Algebras