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Representation Theory: A Homological Algebra Point of View: Algebra and Applications, cartea 19

Autor Alexander Zimmermann
en Limba Engleză Hardback – 26 aug 2014
Introducing the representation theory of groups and finite dimensional algebras, first studying basic non-commutative ring theory, this book covers the necessary background on elementary homological algebra and representations of groups up to block theory. It further discusses vertices, defect groups, Green and Brauer correspondences and Clifford theory. Whenever possible the statements are presented in a general setting for more general algebras, such as symmetric finite dimensional algebras over a field.
Then, abelian and derived categories are introduced in detail and are used to explain stable module categories, as well as derived categories and their main invariants and links between them. Group theoretical applications of these theories are given – such as the structure of blocks of cyclic defect groups – whenever appropriate. Overall, many methods from the representation theory of algebras are introduced.
Representation Theory assumes only the most basic knowledge of linear algebra, groups, rings and fields and guides the reader in the use of categorical equivalences in the representation theory of groups and algebras. As the book is based on lectures, it will be accessible to any graduate student in algebra and can be used for self-study as well as for classroom use.
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Specificații

ISBN-13: 9783319079677
ISBN-10: 3319079670
Pagini: 695
Ilustrații: XX, 707 p. 59 illus.
Dimensiuni: 155 x 235 x 45 mm
Greutate: 1.13 kg
Ediția:2014
Editura: Springer International Publishing
Colecția Springer
Seria Algebra and Applications

Locul publicării:Cham, Switzerland

Public țintă

Research

Cuprins

Rings, Algebras and Modules.- Modular Representations of Finite Groups.- Abelian and Triangulated Categories.- Morita theory.- Stable Module Categories.- Derived Equivalences.

Recenzii

“The focus of this text is the representation theory of associative algebras and the modular representation theory of finite groups, with an emphasis on the interplay between these two fields. … the text at hand is aimed at a beginning graduate student without prior exposure to homological algebra. … Overall, this book is a great repository of theory, developed almost from scratch, with detailed proofs.” (Alex S. Dugas, Mathematical Reviews, May, 2016)
“This book is intended as a text for first year master students who want to specialize on representation theory, more precisely: representations of finite-dimensional algebras and modular group representations, with special emphasis on homological methods.” (Wolfgang Rump, zbMATH 1306.20001, 2015)
“The author’s intent is to provide an obviously very serious ‘introduction to the representation theory of finite groups and finite dimensional algebras via homological algebra.’ … Zimmermann’s book is geared to initiates and serious algebraists aiming at research in the indicated area. It is clearly a labor of love and fine scholarship, and should succeed in providing guidance and instruction in a most interesting and intricate subject.” (Michael Berg, MAA Reviews, October, 2014)

Notă biografică

Alexander Zimmermann works on equivalences between derived module categories, stable module categories, Hochschild cohomology and integral and modular representations of groups.

Textul de pe ultima copertă

 
Introducing the representation theory of groups and finite dimensional algebras, this book first studies basic non-commutative ring theory, covering the necessary background of elementary homological algebra and representations of groups to block theory.
It further discusses vertices, defect groups, Green and Brauer correspondences and Clifford theory. Whenever possible the statements are presented in a general setting for more general algebras, such as symmetric finite dimensional algebras over a field.
Then, abelian and derived categories are introduced in detail and are used to explain stable module categories, as well as derived categories and their main invariants and links between them. Group theoretical applications of these theories are given – such as the structure of blocks of cyclic defect groups – whenever appropriate. Overall, many methods from the representation theory of algebras are introduced.
Representation Theory assumes only the most basic knowledge of linear algebra, groups, rings and fields, and guides the reader in the use of categorical equivalences in the representation theory of groups and algebras. As the book is based on lectures, it will be accessible to any graduate student in algebra and can be used for self-study as well as for classroom use.

Caracteristici

Provides full proofs of key statements in the modular representation theory of groups Contains a coherent treatment and full proofs of the main results on equivalences between derived categories Introduces stable categories and different types of equivalences between them as well as their respective invariants Is completely self-contained and only assumes a basic knowledge of algebra Includes supplementary material: sn.pub/extras