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. 1. The Spectrum Of A Ring -- 2. The Zariski Topology On The Spectrum -- 3. Representable Group Functors -- 4. Hopf Algebras -- 5. Valuations And Larson Orders -- 6. Formal Group Hopf Orders -- 7. Hopf Orders In Kcp -- 8. Hopf Orders In Kcp2 -- 9. Hopf Orders In Kcp3 -- 10. Hopf Orders And Galois Module Theory -- 11. The Class Group Of Hopf Order -- 12. Open Questions And Research Problems. Robert G. Underwood. Includes Bibliographical References (p. 267-270) And Index.

Underwood, Robert G. (Robert Gene)

New York: Springer

Copernicus

Telos

United States, United States of America

MiU

With wide-ranging connections to fields from theoretical physics to computer science, Hopf algebras offer students a glimpse at the applications of abstract mathematics. This book is unique in making this engaging subject accessible to advanced undergraduate and beginning graduate students. After providing a self-contained introduction to group and ring theory, the book thoroughly treats the concept of the spectrum of a ring and the Zariski topology. In this way the student transitions smoothly from basic abstract algebra 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.
Erscheinungsdatum: 30.08.2011

xiv, 273 pages : 24 cm
Includes bibliographical references (pages 267-270) and index

2023-06-28