An Introduction to Hopf Algebras

2011
ISBN: 978-0-387-72765-3
Verlag: Springer

Buch, Englisch, 273 Seiten, HC runder Rücken kaschiert, Format (B × H): 160 mm x 241 mm, Gewicht: 600 g

The study of Hopf algebras spans many fields in mathematics including topology, algebraic geometry, algebraic number theory, Galois module theory, cohomology of groups, and formal groups and has wide-ranging connections to fields from theoretical physics to computer science. This text is unique in making this engaging subject accessible to advanced graduate and beginning graduate students and focuses on applications of Hopf algebras to algebraic number theory and Galois module theory, providing a smooth transition from modern algebra to Hopf algebras.

After providing an introduction to the spectrum of a ring and the Zariski topology, the text treats presheaves, sheaves, and representable group functors. In this way the student transitions smoothly from basic algebraic geometry to Hopf algebras. The importance of Hopf orders is underscored with applications to algebraic number theory, Galois module theory and the theory of formal groups. By the end of the book, readers will be familiar with established results in the field and ready to pose research questions of their own.

An exercise set is included in each of twelve chapters with questions ranging in difficulty. Open problems and research questions are presented in the last chapter. Prerequisites include an understanding of the material on groups, rings, and fields normally covered in a basic course in modern algebra.

Zielgruppe

Graduate

Autoren/Hrsg.

Underwood, Robert G.

Fachgebiete

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

Inhaltsverzeichnis

Preface.- Some Notation.- 1. The Spectrum of a Ring.-2. The Zariski Topology on the Spectrum.-3. Representable Group Functors.-4. Hopf Algebras. -5. Larson Orders.-6. Formal Group Hopf Orders.-7. Hopf Orders in KC_p.-8. Hopf Orders in KC_{p^2}.-9. Hopf Orders in KC_{p^3}.-10. Hopf Orders and Galois Module Theory.-11. The Class Group of a Hopf Order.-12. Open Questions and Research Problems.-Bibliography.-Index.

Über Autor(innen)

Underwood, Robert G.
Robert G. Underwood is an Associate Professor of Mathematics at Auburn University. His research interests include the classification of Hopf algebra orders in group rings and the application of Hopf orders to Galois module theory. Professor Underwood earned his PhD in Mathematics in 1992 from the State University of New York at Albany, and his Masters in Math Education in 1986, also from SUNY Albany.

Robert G. Underwood is an Associate Professor of Mathematics at Auburn University. His research interests include the classification of Hopf algebra orders in group rings and the application of Hopf orders to Galois module theory. Professor Underwood earned his PhD in Mathematics in 1992 from the State University of New York at Albany, and his Masters in Math Education in 1986, also from SUNY Albany.