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

1. Auflage 2018
ISBN: 978-3-319-98136-9
Verlag: Springer International Publishing

Buch, Englisch, Band 2226, 165 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 283 g

Reihe: Lecture Notes in Mathematics

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.

Zielgruppe

Research

Autoren/Hrsg.

Böhm, Gabriella

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Homologische Algebra
• Mathematik | Informatik Mathematik Algebra Algebraische Strukturen, Gruppentheorie

Inhaltsverzeichnis

- Introduction. - Lifting to Eilenberg–Moore Categories. - (Hopf) Bimonads. - (Hopf) Bialgebras. - (Hopf) Bialgebroids. - Weak (Hopf) Bialgebras. - (Hopf) Bimonoids in Duoidal Categories. - Solutions to the Exercises.

Über Autor(innen)

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.