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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.
List of contents
Rings, Algebras and Modules.- Modular Representations of Finite Groups.- Abelian and Triangulated Categories.- Morita theory.- Stable Module Categories.- Derived Equivalences.
About the author
Alexander Zimmermann works on equivalences between derived module categories, stable module categories, Hochschild cohomology and integral and modular representations of groups.
Summary
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.
Additional text
“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)
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"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)
