From Rings and Modules to Hopf Algebras: One Flew Over the Algebraist's Nest
Broué, Michel
- 出版商: Springer
- 出版日期: 2024-01-31
- 售價: $3,730
- 貴賓價: 9.5 折 $3,544
- 語言: 英文
- 頁數: 533
- 裝訂: Hardcover - also called cloth, retail trade, or trade
- ISBN: 303150061X
- ISBN-13: 9783031500619
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商品描述
This textbook provides an introduction to fundamental concepts of algebra at upper undergraduate to graduate level, covering the theory of rings, fields and modules, as well as the representation theory of finite groups.
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.
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.
商品描述(中文翻譯)
這本教科書提供了一個介紹代數基本概念的入門,適用於高年級本科生到研究生水平,涵蓋了環、域和模的理論,以及有限群的表示理論。
整本書的講解都依賴於通用的構造,系統地使用商集和范畴論,這些語言在第一章中介紹。該書分為四個部分。第一部分和第二部分涵蓋了環和模的基礎,域論和有限群表示的一般性,並強調多項式環及其理想。第三部分以戴德金環上有限生成模的結構理論為高潮,並應用於阿貝爾群、線性映射和代數數論的基礎。第四部分是對特徵為零的域上有限群的線性表示的廣泛研究,包括分級表示和分級特徵,以及關於Drinfeld-Lusztig雙群代數的最後一章,這是首次在這個水平的教科書中出現。
本書基於作者在教授各種代數方面的二十多年經驗,主要在巴黎高等師範學校和北京大學進行教學,反映了作者課程的受眾。特別是,假設讀者具備抽象代數的基礎,如線性代數和初等群論。每個部分都可以用於一門課程,並附帶一些臨時的補充關於范畴語言。由於該書選題豐富,也可以作為學生從本科到高級研究生階段的參考書。
作者簡介
Michel Broué is a distinguished French mathematician specialising in algebra. He is known for his work on modular representations of finite groups (where a famous conjecture bears his name), on the "generic approach" to representations of finite reductive groups, as well as for work on related areas like cyclotomic Hecke algebras. During a career spanning half a century, he has been the editor-in-chief of the Journal of Algebra (2001-2017), the director of the Department of Mathematics and Computer Science of the École Normale Supérieure and the director of the Institut Henri Poincaré (1999-2009).
作者簡介(中文翻譯)
Michel Broué是一位傑出的法國數學家,專攻代數學。他以他在有限群的模表示方面的工作而聞名(其中一個著名的猜想以他的名字命名),以及對有限紅化群表示的“通用方法”以及類似領域(如循環Hecke代數)的工作。在長達半個世紀的職業生涯中,他曾擔任《代數學雜誌》的主編(2001-2017),法國高等師範學校數學與計算機科學系的主任以及亨利·庞加莱研究所的主任(1999-2009)。