Lecture Notes on Cluster Algebras PDF Download
Are you looking for read ebook online? Search for your book and save it on your Kindle device, PC, phones or tablets. Download Lecture Notes on Cluster Algebras PDF full book. Access full book title Lecture Notes on Cluster Algebras by Robert J. Marsh. Download full books in PDF and EPUB format.
Author: Robert J. Marsh
Publisher: Erich Schmidt Verlag GmbH & Co. KG
ISBN: 9783037191309
Category : Mathematics
Languages : en
Pages : 132
Get Book Here
Book Description
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: Robert J. Marsh
Publisher: Erich Schmidt Verlag GmbH & Co. KG
ISBN: 9783037191309
Category : Mathematics
Languages : en
Pages : 132
Get Book Here
Book Description
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: Sergey Fomin
Publisher: American Mathematical Soc.
ISBN: 1470429675
Category : Mathematics
Languages : en
Pages : 110
Get Book Here
Book Description
For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, the authors construct a geometric realization in terms of suitable decorated Teichmüller space of the surface. On the geometric side, this requires opening the surface at each interior marked point into an additional geodesic boundary component. On the algebraic side, it relies on the notion of a non-normalized cluster algebra and the machinery of tropical lambda lengths. The authors' model allows for an arbitrary choice of coefficients which translates into a choice of a family of integral laminations on the surface. It provides an intrinsic interpretation of cluster variables as renormalized lambda lengths of arcs on the surface. Exchange relations are written in terms of the shear coordinates of the laminations and are interpreted as generalized Ptolemy relations for lambda lengths. This approach gives alternative proofs for the main structural results from the authors' previous paper, removing unnecessary assumptions on the surface.
Author: Ralf Schiffler
Publisher: Springer
ISBN: 3319092049
Category : Mathematics
Languages : en
Pages : 233
Get Book Here
Book Description
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
Publisher: American Mathematical Soc.
ISBN: 0821849727
Category : Mathematics
Languages : en
Pages : 264
Get Book Here
Book Description
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: Ibrahim Assem
Publisher: Springer
ISBN: 3319745859
Category : Mathematics
Languages : en
Pages : 231
Get Book Here
Book Description
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: Lidia Angeleri Hügel
Publisher: Cambridge University Press
ISBN: 9780521680455
Category : Mathematics
Languages : en
Pages : 482
Get Book Here
Book Description
A handbook of key articles providing both an introduction and reference for newcomers and experts alike.
Author: Anders Björner
Publisher: Cambridge University Press
ISBN: 052177750X
Category : Mathematics
Languages : en
Pages : 564
Get Book Here
Book Description
First comprehensive, accessible account; second edition has expanded bibliography and a new appendix surveying recent research.
Author: Thorsten Holm
Publisher: Cambridge University Press
ISBN: 1139488880
Category : Mathematics
Languages : en
Pages : 473
Get Book Here
Book Description
A 2010 collection of survey articles by leading experts covering fundamental aspects of triangulated categories, as well as applications in algebraic geometry, representation theory, commutative algebra, microlocal analysis and algebraic topology. This is a valuable reference for experts and a useful introduction for graduate students entering the field.
Author: Graham J. Leuschke
Publisher: American Mathematical Soc.
ISBN: 1470435764
Category : Mathematics
Languages : en
Pages : 294
Get Book Here
Book Description
Contains the proceedings of the 17th Workshop and International Conference on Representations of Algebras (ICRA 2016), held in August 2016, at Syracuse University. This volume includes 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.
Author: Kevin Costello
Publisher: Cambridge University Press
ISBN: 1107163102
Category : Mathematics
Languages : en
Pages : 399
Get Book Here
Book Description
This first volume develops factorization algebras with a focus upon examples exhibiting their use in field theory, which will be useful for researchers and graduates.
