Derived Equivalences for Group Rings
Derived Equivalences for Group Rings

Author: Steffen König

Publisher: Springer

Published: 2006-11-14

Total Pages: 256

ISBN-13: 3540697489

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A self-contained introduction is given to J. Rickard's Morita theory for derived module categories and its recent applications in representation theory of finite groups. In particular, Broué's conjecture is discussed, giving a structural explanation for relations between the p-modular character table of a finite group and that of its "p-local structure". The book is addressed to researchers or graduate students and can serve as material for a seminar. It surveys the current state of the field, and it also provides a "user's guide" to derived equivalences and tilting complexes. Results and proofs are presented in the generality needed for group theoretic applications.

The Block Theory of Finite Group Algebras:
The Block Theory of Finite Group Algebras:

Author: Markus Linckelmann

Publisher: Cambridge University Press

Published: 2018-05-24

Total Pages: 528

ISBN-13: 1108642187

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This is a comprehensive introduction to the modular representation theory of finite groups, with an emphasis on block theory. The two volumes take into account classical results and concepts as well as some of the modern developments in the area. Volume 1 introduces the broader context, starting with general properties of finite group algebras over commutative rings, moving on to some basics in character theory and the structure theory of algebras over complete discrete valuation rings. In Volume 2, blocks of finite group algebras over complete p-local rings take centre stage, and many key results which have not appeared in a book before are treated in detail. In order to illustrate the wide range of techniques in block theory, the book concludes with chapters classifying the source algebras of blocks with cyclic and Klein four defect groups, and relating these classifications to the open conjectures that drive block theory.

The Block Theory of Finite Group Algebras: Volume 1
The Block Theory of Finite Group Algebras: Volume 1

Author: Markus Linckelmann

Publisher: Cambridge University Press

Published: 2018-05-24

Total Pages: 527

ISBN-13: 1108575315

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This is a comprehensive introduction to the modular representation theory of finite groups, with an emphasis on block theory. The two volumes take into account classical results and concepts as well as some of the modern developments in the area. Volume 1 introduces the broader context, starting with general properties of finite group algebras over commutative rings, moving on to some basics in character theory and the structure theory of algebras over complete discrete valuation rings. In Volume 2, blocks of finite group algebras over complete p-local rings take centre stage, and many key results which have not appeared in a book before are treated in detail. In order to illustrate the wide range of techniques in block theory, the book concludes with chapters classifying the source algebras of blocks with cyclic and Klein four defect groups, and relating these classifications to the open conjectures that drive block theory.

Homological Methods, Representation Theory, and Cluster Algebras
Homological Methods, Representation Theory, and Cluster Algebras

Author: Ibrahim Assem

Publisher: Springer

Published: 2018-04-18

Total Pages: 231

ISBN-13: 3319745859

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This text presents six mini-courses, all devoted to interactions between representation theory of algebras, homological algebra, and the new ever-expanding theory of cluster algebras. The interplay between the topics discussed in this text will continue to grow and this collection of courses stands as a partial testimony to this new development. The courses are useful for any mathematician who would like to learn more about this rapidly developing field; the primary aim is to engage graduate students and young researchers. Prerequisites include knowledge of some noncommutative algebra or homological algebra. Homological algebra has always been considered as one of the main tools in the study of finite-dimensional algebras. The strong relationship with cluster algebras is more recent and has quickly established itself as one of the important highlights of today’s mathematical landscape. This connection has been fruitful to both areas—representation theory provides a categorification of cluster algebras, while the study of cluster algebras provides representation theory with new objects of study. The six mini-courses comprising this text were delivered March 7–18, 2016 at a CIMPA (Centre International de Mathématiques Pures et Appliquées) research school held at the Universidad Nacional de Mar del Plata, Argentina. This research school was dedicated to the founder of the Argentinian research group in representation theory, M.I. Platzeck. The courses held were: Advanced homological algebra Introduction to the representation theory of algebras Auslander-Reiten theory for algebras of infinite representation type Cluster algebras arising from surfaces Cluster tilted algebras Cluster characters Introduction to K-theory Brauer graph algebras and applications to cluster algebras

Handbook of Tilting Theory
Handbook of Tilting Theory

Author: Lidia Angeleri Hügel

Publisher: Cambridge University Press

Published: 2007-01-04

Total Pages: 482

ISBN-13: 9780521680455

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A handbook of key articles providing both an introduction and reference for newcomers and experts alike.

The Block Theory of Finite Group Algebras: Volume 2
The Block Theory of Finite Group Algebras: Volume 2

Author: Markus Linckelmann

Publisher: Cambridge University Press

Published: 2018-05-24

Total Pages: 523

ISBN-13: 1108562582

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This is a comprehensive introduction to the modular representation theory of finite groups, with an emphasis on block theory. The two volumes take into account classical results and concepts as well as some of the modern developments in the area. Volume 1 introduces the broader context, starting with general properties of finite group algebras over commutative rings, moving on to some basics in character theory and the structure theory of algebras over complete discrete valuation rings. In Volume 2, blocks of finite group algebras over complete p-local rings take centre stage, and many key results which have not appeared in a book before are treated in detail. In order to illustrate the wide range of techniques in block theory, the book concludes with chapters classifying the source algebras of blocks with cyclic and Klein four defect groups, and relating these classifications to the open conjectures that drive block theory.

Representations of Reductive Groups
Representations of Reductive Groups

Author: Roger W. Carter

Publisher: Cambridge University Press

Published: 1998-09-03

Total Pages: 203

ISBN-13: 0521643252

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This volume provides a very accessible introduction to the representation theory of reductive algebraic groups.

Mathematisches Institut Georg-august-universität Göttingen, Seminars Summer Term 2004
Mathematisches Institut Georg-august-universität Göttingen, Seminars Summer Term 2004

Author: Yuri Tschinkel

Publisher: Universitätsverlag Göttingen

Published: 2004

Total Pages: 200

ISBN-13: 3930457709

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This volume contains lecture notes from the seminars [alpha]Number Theory", [alpha]Algebraic Geometry" and [alpha]Geometric methods in representation theory" which took place at the Mathematics Institute of the University of Göttingen during the Summer Term 2004. Most contributions report on recent work by the authors.

Algebra - Representation Theory
Algebra - Representation Theory

Author: Klaus W. Roggenkamp

Publisher: Springer Science & Business Media

Published: 2001-08-31

Total Pages: 488

ISBN-13: 9780792371137

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Over the last three decades representation theory of groups, Lie algebras and associative algebras has undergone a rapid development through the powerful tool of almost split sequences and the Auslander-Reiten quiver. Further insight into the homology of finite groups has illuminated their representation theory. The study of Hopf algebras and non-commutative geometry is another new branch of representation theory which pushes the classical theory further. All this can only be seen in connection with an understanding of the structure of special classes of rings. The aim of this book is to introduce the reader to some modern developments in: Lie algebras, quantum groups, Hopf algebras and algebraic groups; non-commutative algebraic geometry; representation theory of finite groups and cohomology; the structure of special classes of rings.

Structured Ring Spectra
Structured Ring Spectra

Author: Andrew Baker

Publisher: Cambridge University Press

Published: 2004-11-18

Total Pages: 246

ISBN-13: 9780521603058

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This book contains some important new contributions to the theory of structured ring spectra.