Advertisement

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

  • Gabriella Böhm

Part of the Lecture Notes in Mathematics book series (LNM, volume 2226)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Gabriella Böhm
    Pages 1-6
  3. Gabriella Böhm
    Pages 7-28
  4. Gabriella Böhm
    Pages 29-46
  5. Gabriella Böhm
    Pages 47-58
  6. Gabriella Böhm
    Pages 59-73
  7. Gabriella Böhm
    Pages 75-97
  8. Gabriella Böhm
    Pages 99-123
  9. Gabriella Böhm
    Pages 125-154
  10. Back Matter
    Pages 155-165

About this book

Introduction

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.

Keywords

MSC (2010): 16T10, 16T05, 18C15, 18D10, 18D15 bialgebra Hopf algebra bialgebroid Hopf algebroid weak bialgebra weak Hopf algebra bimonad Hopf monad monoidal category duoidal category Eilenberg-Moore lifting

Authors and affiliations

  • Gabriella Böhm
    • 1
  1. 1.Wigner Research Centre for PhysicsHungarian Academy of SciencesBudapestHungary

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-98137-6
  • Copyright Information Springer Nature Switzerland AG 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-98136-9
  • Online ISBN 978-3-319-98137-6
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • Buy this book on publisher's site