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An Introduction to Hopf Algebras / by Robert G. Underwood
Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: SpringerLink BücherPublisher: New York, NY : Springer Science+Business Media, LLC, 2011Description: Online-Ressource (XIV, 273p. 3 illus, digital)ISBN:- 9780387727660
- 512.55
- 512.55 512/.55
- 512
- QA150-272
- QA613.8
Contents:
Summary: 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.Summary: 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.PPN: PPN: 1651019916Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
""An Introduction to Hopf Algebras""; ""Preface""; ""Contents""; ""Notation""; ""Chapter 1: The Spectrum of a Ring ""; ""1.1 Introduction to the Spectrum""; ""1.2 The Associated Map of Spectra""; ""1.3 Nilpotent Elements""; ""1.4 Chapter Exercises""; ""Chapter 2: The Zariski Topology on the Spectrum ""; ""2.1 Some Topology""; ""2.2 Basis for a Topological Space""; ""2.3 Sheaves""; ""2.4 Representable Functors""; ""2.5 Chapter Exercises""; ""Chapter 3: Representable Group Functors ""; ""3.1 Introduction to Representable Group Functors""; ""3.2 Homomorphisms of R-Group Schemes""
""3.3 Short Exact Sequences""""3.4 An Example""; ""3.5 Chapter Exercises""; ""Chapter 4: Hopf Algebras ""; ""4.1 Introduction to Hopf Algebras""; ""4.2 Dedekind Domains""; ""4.3 Hopf Modules""; ""4.4 Hopf Orders""; ""4.5 Chapter Exercises""; ""Chapter 5: Valuations and Larson Orders ""; ""5.1 Valuations""; ""5.2 Group Valuations""; ""5.3 Larson Orders""; ""5.4 Chapter Exercises""; ""Chapter 6: Formal Group Hopf Orders ""; ""6.1 Formal Groups""; ""6.2 Formal Group Hopf Orders""; ""6.3 Chapter Exercises""; ""Chapter 7: Hopf Orders in KCp ""; ""7.1 Classification of Hopf Orders in KCp""
""7.2 Chapter Exercises""""Chapter 8: Hopf Orders in KCp2 ""; ""8.1 The Valuation Condition""; ""8.2 Some Cohomology""; ""8.3 Greither Orders""; ""8.4 Hopf Orders in KC4, KC9""; ""8.5 Chapter Exercises""; ""Chapter 9: Hopf Orders in KCp3 ""; ""9.1 Duality Hopf Orders in KCp3""; ""9.2 Circulant Matrices and Hopf Orders in KCp3""; ""9.3 Chapter Exercises""; ""Chapter 10: Hopf Orders and Galois Module Theory ""; ""10.1 Some Galois Theory""; ""10.2 Ramification""; ""10.3 Galois Extensions of Rings""; ""10.4 Hopf-Galois Extensions of a Local Ring""; ""10.5 The Normal Basis Theorem""
""10.6 Chapter Exercises""""Chapter 11: The Class Group of a Hopf Order ""; ""11.1 The Class Group of a Number Field""; ""11.2 The Class Group of a Hopf Order""; ""11.3 The Hopf-Swan Subgroup""; ""11.4 Chapter Exercises""; ""Chapter 12: Open Questions and Research Problems ""; ""12.1 The Spectrum of a Ring""; ""12.2 Hopf Algebras""; ""12.3 Valuations and Larson Orders""; ""12.4 Hopf Orders in KCp2""; ""12.5 Hopf Orders in KCp3""; ""12.6 Hopf Orders and Galois Module Theory""; ""12.7 The Class Group of a Hopf Order""; ""Bibliography""; ""Index""
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