Homological Methods Representation Theory And Cluster Algebras

Author : Ibrahim Assem,Sonia Trepode
Publisher : Springer
Page : 223 pages
File Size : 47,9 Mb
Release : 2018-04-18
Category : Mathematics
ISBN : 9783319745855

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Homological Methods, Representation Theory, and Cluster Algebras by Ibrahim Assem,Sonia Trepode Pdf

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

Author : Ibrahim Assem,Christof Geiß,Sonia Trepode
Publisher : American Mathematical Soc.
Page : 257 pages
File Size : 43,7 Mb
Release : 2021-01-06
Category : Education
ISBN : 9781470451592

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Advances in Representation Theory of Algebras by Ibrahim Assem,Christof Geiß,Sonia Trepode Pdf

The Seventh ARTA (“Advances in Representation Theory of Algebras VII”) conference took place at the Instituto de Matemáticas of the Universidad Nacional Autónoma de México, in Mexico City, from September 24–28, 2018, in honor of José Antonio de la Peña's 60th birthday. Papers in this volume cover topics Professor de la Peña worked on, such as covering theory, tame algebras, and the use of quadratic forms in representation theory. Also included are papers on the categorical approach to representations of algebras and relations to Lie theory, Cohen–Macaulay modules, quantum groups and other algebraic structures.

Author : David Jordan,Nadia Mazza,Sibylle Schroll
Publisher : Cambridge University Press
Page : 408 pages
File Size : 51,6 Mb
Release : 2023-07-31
Category : Mathematics
ISBN : 9781009103473

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Modern Trends in Algebra and Representation Theory by David Jordan,Nadia Mazza,Sibylle Schroll Pdf

Expanding upon the material delivered during the LMS Autumn Algebra School 2020, this volume reflects the fruitful connections between different aspects of representation theory. Each survey article addresses a specific subject from a modern angle, beginning with an exploration of the representation theory of associative algebras, followed by the coverage of important developments in Lie theory in the past two decades, before the final sections introduce the reader to three strikingly different aspects of group theory. Written at a level suitable for graduate students and researchers in related fields, this book provides pure mathematicians with a springboard into the vast and growing literature in each area.

Author : Anonim
Publisher : Springer
Page : 244 pages
File Size : 47,8 Mb
Release : 2011-10-20
Category : Electronic
ISBN : 0817682759

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Highlights in Lie Algebraic Methods by Anonim Pdf

Author : Robert J. Marsh
Publisher : Erich Schmidt Verlag GmbH & Co. KG
Page : 132 pages
File Size : 53,7 Mb
Release : 2013
Category : Cluster algebras
ISBN : 3037191309

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Lecture Notes on Cluster Algebras by Robert J. Marsh Pdf

Cluster algebras are combinatorially defined commutative algebras which were introduced by S. Fomin and A. Zelevinsky as a tool for studying the dual canonical basis of a quantized enveloping algebra and totally positive matrices. The aim of these notes is to give an introduction to cluster algebras which is accessible to graduate students or researchers interested in learning more about the field while giving a taste of the wide connections between cluster algebras and other areas of mathematics. The approach taken emphasizes combinatorial and geometric aspects of cluster algebras. Cluster algebras of finite type are classified by the Dynkin diagrams, so a short introduction to reflection groups is given in order to describe this and the corresponding generalized associahedra. A discussion of cluster algebra periodicity, which has a close relationship with discrete integrable systems, is included. This book ends with a description of the cluster algebras of finite mutation type and the cluster structure of the homogeneous coordinate ring of the Grassmannian, both of which have a beautiful description in terms of combinatorial geometry.

Author : Graham J. Leuschke,Frauke Bleher,Ralf Schiffler,Dan Zacharia
Publisher : American Mathematical Soc.
Page : 296 pages
File Size : 55,9 Mb
Release : 2018
Category : Algebra
ISBN : 9781470435769

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Representations of Algebras by Graham J. Leuschke,Frauke Bleher,Ralf Schiffler,Dan Zacharia Pdf

This volume contains the proceedings of the 17th Workshop and International Conference on Representations of Algebras (ICRA 2016), held from August 10-19, 2016, at Syracuse University, Syracuse, NY. Included are three survey articles based on short courses in the areas of commutative algebraic groups, modular group representation theory, and thick tensor ideals of bounded derived categories. Other articles represent contributions to areas in and related to representation theory, such as noncommutative resolutions, twisted commutative algebras, and upper cluster algebras.

Author : Anthony Joseph,Anna Melnikov,Ivan Penkov
Publisher : Birkhäuser
Page : 227 pages
File Size : 45,6 Mb
Release : 2011-10-19
Category : Mathematics
ISBN : 0817682732

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Highlights in Lie Algebraic Methods by Anthony Joseph,Anna Melnikov,Ivan Penkov Pdf

This volume consists of expository and research articles that highlight the various Lie algebraic methods used in mathematical research today. Key topics discussed include spherical varieties, Littelmann Paths and Kac–Moody Lie algebras, modular representations, primitive ideals, representation theory of Artin algebras and quivers, Kac–Moody superalgebras, categories of Harish–Chandra modules, cohomological methods, and cluster algebras.

Author : Dolors Herbera,Wolfgang Pitsch,Santiago Zarzuela
Publisher : Birkhäuser
Page : 0 pages
File Size : 45,7 Mb
Release : 2016-12-06
Category : Mathematics
ISBN : 3319454404

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Extended Abstracts Spring 2015 by Dolors Herbera,Wolfgang Pitsch,Santiago Zarzuela Pdf

This book includes 33 expanded abstracts of selected talks given at the two workshops "Homological Bonds Between Commutative Algebra and Representation Theory" and "Brave New Algebra: Opening Perspectives," and the conference "Opening Perspectives in Algebra, Representations, and Topology," held at the Centre de Recerca Matemàtica (CRM) in Barcelona between January and June 2015. These activities were part of the one-semester intensive research program "Interactions Between Representation Theory, Algebraic Topology and Commutative Algebra (IRTATCA)." Most of the abstracts present preliminary versions of not-yet published results and cover a large number of topics (including commutative and non commutative algebra, algebraic topology, singularity theory, triangulated categories, representation theory) overlapping with homological methods. This comprehensive book is a valuable resource for the community of researchers interested in homological algebra in a broad sense, and those curious to learn the latest developments in the area. It appeals to established researchers as well as PhD and postdoctoral students who want to learn more about the latest advances in these highly active fields of research.

Author : Ralf Schiffler
Publisher : Springer
Page : 230 pages
File Size : 42,8 Mb
Release : 2014-09-04
Category : Mathematics
ISBN : 9783319092041

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Quiver Representations by Ralf Schiffler Pdf

This book is intended to serve as a textbook for a course in Representation Theory of Algebras at the beginning graduate level. The text has two parts. In Part I, the theory is studied in an elementary way using quivers and their representations. This is a very hands-on approach and requires only basic knowledge of linear algebra. The main tool for describing the representation theory of a finite-dimensional algebra is its Auslander-Reiten quiver, and the text introduces these quivers as early as possible. Part II then uses the language of algebras and modules to build on the material developed before. The equivalence of the two approaches is proved in the text. The last chapter gives a proof of Gabriel’s Theorem. The language of category theory is developed along the way as needed.

Author : Michael Gekhtman,Michael Shapiro,Alek Vainshtein
Publisher : American Mathematical Soc.
Page : 264 pages
File Size : 44,5 Mb
Release : 2010
Category : Cluster algebras
ISBN : 9780821849729

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Cluster Algebras and Poisson Geometry by Michael Gekhtman,Michael Shapiro,Alek Vainshtein Pdf

The first book devoted to cluster algebras, this work contains chapters on Poisson geometry and Schubert varieties; an introduction to cluster algebras and their main properties; and geometric aspects of the cluster algebra theory, in particular on its relations to Poisson geometry and to the theory of integrable systems.

Author : César Polcino Milies
Publisher : American Mathematical Soc.
Page : 336 pages
File Size : 44,6 Mb
Release : 2011
Category : Associative algebras
ISBN : 9780821852392

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Groups, Algebras and Applications by César Polcino Milies Pdf

Contains the proceedings of the XVIII Latin American Algebra Colloquium, held from August 3-8, 2009, in Sao Paulo, Brazil. It includes research articles as well as up-to-date surveys covering several directions of current research in algebra, such as Asymptotic Codimension Growth, Hopf Algebras, Structure Theory of both Associative and Non-Associative Algebras, Partial Actions of Groups on Rings, and contributions to Coding Theory.

Author : Harm Derksen,Jerzy Weyman
Publisher : American Mathematical Soc.
Page : 344 pages
File Size : 49,9 Mb
Release : 2017-11-29
Category : Directed graphs
ISBN : 9781470425562

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An Introduction to Quiver Representations by Harm Derksen,Jerzy Weyman Pdf

This book is an introduction to the representation theory of quivers and finite dimensional algebras. It gives a thorough and modern treatment of the algebraic approach based on Auslander-Reiten theory as well as the approach based on geometric invariant theory. The material in the opening chapters is developed starting slowly with topics such as homological algebra, Morita equivalence, and Gabriel's theorem. Next, the book presents Auslander-Reiten theory, including almost split sequences and the Auslander-Reiten transform, and gives a proof of Kac's generalization of Gabriel's theorem. Once this basic material is established, the book goes on with developing the geometric invariant theory of quiver representations. The book features the exposition of the saturation theorem for semi-invariants of quiver representations and its application to Littlewood-Richardson coefficients. In the final chapters, the book exposes tilting modules, exceptional sequences and a connection to cluster categories. The book is suitable for a graduate course in quiver representations and has numerous exercises and examples throughout the text. The book will also be of use to experts in such areas as representation theory, invariant theory and algebraic geometry, who want to learn about applications of quiver representations to their fields.

Author : Alexander Zimmermann
Publisher : Springer
Page : 707 pages
File Size : 51,5 Mb
Release : 2014-08-15
Category : Mathematics
ISBN : 9783319079684

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Representation Theory by Alexander Zimmermann Pdf

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.

Author : Andrzej Skowroński
Publisher : European Mathematical Society
Page : 732 pages
File Size : 54,7 Mb
Release : 2008
Category : Representations of algebras
ISBN : 3037190620

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Trends in Representation Theory of Algebras and Related Topics by Andrzej Skowroński Pdf

This book is concerned with recent trends in the representation theory of algebras and its exciting interaction with geometry, topology, commutative algebra, Lie algebras, quantum groups, homological algebra, invariant theory, combinatorics, model theory and theoretical physics. The collection of articles, written by leading researchers in the field, is conceived as a sort of handbook providing easy access to the present state of knowledge and stimulating further development. The topics under discussion include diagram algebras, Brauer algebras, cellular algebras, quasi-hereditary algebras, Hall algebras, Hecke algebras, symplectic reflection algebras, Cherednik algebras, Kashiwara crystals, Fock spaces, preprojective algebras, cluster algebras, rank varieties, varieties of algebras and modules, moduli of representations of quivers, semi-invariants of quivers, Cohen-Macaulay modules, singularities, coherent sheaves, derived categories, spectral representation theory, Coxeter polynomials, Auslander-Reiten theory, Calabi-Yau triangulated categories, Poincare duality spaces, selfinjective algebras, periodic algebras, stable module categories, Hochschild cohomologies, deformations of algebras, Galois coverings of algebras, tilting theory, algebras of small homological dimensions, representation types of algebras, and model theory. This book consists of fifteen self-contained expository survey articles and is addressed to researchers and graduate students in algebra as well as a broader mathematical community. They contain a large number of open problems and give new perspectives for research in the field.

Author : Martin Herschend,Osamu Iyama,Hiroyuki Minamoto,Steffen Oppermann
Publisher : American Mathematical Society
Page : 156 pages
File Size : 53,6 Mb
Release : 2023-05-23
Category : Mathematics
ISBN : 9781470456313

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Representation Theory of Geigle-Lenzing Complete Intersections by Martin Herschend,Osamu Iyama,Hiroyuki Minamoto,Steffen Oppermann Pdf

View the abstract. https://www.ams.org/bookstore/pspdf/memo-285-1412-abstract.pdf?