Overview
A highly original introduction to basic tools of algebra, from a categorical point of view
Includes advanced material on representation theory such as the Drinfeld–Lusztig double
Unusually rich in examples and exercises
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Table of contents (18 chapters)
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Front Matter
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Rings and Modules
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Front Matter
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Integral Domains, Polynomials, Fields
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Front Matter
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Finitely Generated Modules
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Front Matter
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Characteristic Zero Linear Representations of Finite Groups
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Front Matter
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Keywords
- Polynomial rings
- Ideals
- Principal ideal domain
- Dedekind ring
- Modules
- Discriminant
- Resultant
- Representations of finite groups
- Burnside’s marks
- Projective representations
- Graded representations
- Graded characters
- Drinfeld-Lusztig double
- Verlinde formula
- Galois theory
- Nullstellensatz
- pseudo-bases
- Artin's theorem
- Brauer's theorem
- Monoidal category
About this book
Throughout the book, the exposition relies on universal constructions, making systematic use of quotients and category theory — whose language is introduced in the first chapter. The book is divided into four parts. Parts I and II cover foundations of rings and modules, field theory and generalities on finite group representations, insisting on rings of polynomials and their ideals. Part III culminates in the structure theory of finitely generated modules over Dedekind domains and its applications to abelian groups, linear maps, and foundations of algebraic number theory. Part IV is an extensive study of linear representations of finite groups over fields of characteristic zero, including graded representations and graded characters as well as a final chapter on the Drinfeld–Lusztig double of a group algebra, appearing for the first time in a textbook at this level.
Based on over twenty years of teaching various aspects of algebra, mainly at the École Normale Supérieure (Paris) and at Peking University, the book reflects the audiences of the author's courses. In particular, foundations of abstract algebra, like linear algebra and elementary group theory, are assumed of the reader. Each of the of four parts can be used for a course — with a little ad hoc complement on the language of categories. Thanks to its rich choice of topics, the book can also serve students as a reference throughout their studies, from undergraduate to advanced graduate level.
Authors and Affiliations
About the author
Bibliographic Information
Book Title: From Rings and Modules to Hopf Algebras
Book Subtitle: One Flew Over the Algebraist's Nest
Authors: Michel Broué
DOI: https://doi.org/10.1007/978-3-031-50062-6
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024
Hardcover ISBN: 978-3-031-50061-9Published: 31 January 2024
eBook ISBN: 978-3-031-50062-6Published: 30 January 2024
Edition Number: 1
Number of Pages: X, 533
Number of Illustrations: 10 b/w illustrations, 10 illustrations in colour
Topics: Algebra, Field Theory and Polynomials, Group Theory and Generalizations, Category Theory, Homological Algebra, Commutative Rings and Algebras