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Author: TOMOKI
Publisher:
ISBN: 9784864971058
Category : Mathematics
Languages : en
Pages : 0
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Book Description
The theme of this monograph is the relation between cluster algebras and scattering diagrams. Cluster algebras were introduced by Fomin and Zelevinsky around 2000 as an algebraic and combinatorial structure originated in Lie theory. Recently, Gross, Hacking, Keel, and Kontsevich solved several important conjectures in cluster algebra theory by the scattering diagram method introduced in the homological mirror symmetry. This monograph is the first comprehensive exposition of this important development. The text consists of three parts. Part I is a first step guide to the theory of cluster algebras for readers without any knowledge on cluster algebras. Part II is the main part of the monograph, where we focus on the column sign-coherence of C-matrices and the Laurent positivity for cluster patterns, both of which were conjectured by Fomin and Zelevinsky and proved by Gross, Hacking, Keel, and Kontsevich based on the scattering diagram method. Part III is a self-contained exposition of several fundamental properties of cluster scattering diagrams with emphasis on the roles of the dilogarithm elements and the pentagon relation. As a specific feature of this monograph, each part is written without explicitly relying on the other parts. Thus, readers can start reading from any part depending on their interest and knowledge.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets
Author: TOMOKI
Publisher:
ISBN: 9784864971058
Category : Mathematics
Languages : en
Pages : 0
Get Book Here
Book Description
The theme of this monograph is the relation between cluster algebras and scattering diagrams. Cluster algebras were introduced by Fomin and Zelevinsky around 2000 as an algebraic and combinatorial structure originated in Lie theory. Recently, Gross, Hacking, Keel, and Kontsevich solved several important conjectures in cluster algebra theory by the scattering diagram method introduced in the homological mirror symmetry. This monograph is the first comprehensive exposition of this important development. The text consists of three parts. Part I is a first step guide to the theory of cluster algebras for readers without any knowledge on cluster algebras. Part II is the main part of the monograph, where we focus on the column sign-coherence of C-matrices and the Laurent positivity for cluster patterns, both of which were conjectured by Fomin and Zelevinsky and proved by Gross, Hacking, Keel, and Kontsevich based on the scattering diagram method. Part III is a self-contained exposition of several fundamental properties of cluster scattering diagrams with emphasis on the roles of the dilogarithm elements and the pentagon relation. As a specific feature of this monograph, each part is written without explicitly relying on the other parts. Thus, readers can start reading from any part depending on their interest and knowledge.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets
Author: Nakanishi
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Author: Lang Mou
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
In this dissertation, we study the phenomenon of wall-crossing structures in cluster algebras of Fomin and Zelevinsky, with examples including cluster scattering diagrams of Gross, Hacking, Keel, and Kontsevich (GHKK) and stability scattering diagrams of Bridgeland. We show that in general, every consistent scattering diagram admits a canonical underlying cone complex structure. We describe mutations of the stability scattering diagram of a quiver with non-degenerate potential. Then we use this description to prove that the stability scattering diagram admits the so-called cluster complex structure. As a consequence, we verify if a quiver admits a reddening sequence, a conjecture of Kontsevich and Soibelman that the associated cluster scattering diagram is equivalent to the stability scattering diagram of the same quiver with a non-degenerate potential. We also give another proof of the Caldero-Chapoton formula of cluster monomials using scattering diagrams. Skew-symmetrizable cluster algebras need extra care. We define a Langlands dual version of the cluster scattering diagram of GHKK and show that it admits a cluster complex structure that is Langlands dual to GHKK's version. We use it to describe the cluster monomials of skew- symmetrizable cluster algebras in terms of theta functions. Then we study the Hall algebra scattering diagram associated to the Geiss-Leclerc-Schröer algebra of an acyclic skew-symmetrizable matrix with a skew-symmetrizer. We show that it admits the same cluster complex structure as the aforementioned Langlands dual cluster scattering diagram. In the end, we extend the theory of scattering diagrams to Chekhov and Shapiro's generalized cluster algebras.
Author: Robert J. Marsh
Publisher: Erich Schmidt Verlag GmbH & Co. KG
ISBN: 9783037191309
Category : Cluster algebras
Languages : en
Pages : 132
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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: Michael Gekhtman
Publisher: American Mathematical Soc.
ISBN: 0821849727
Category : Mathematics
Languages : en
Pages : 264
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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: Sergey Fomin
Publisher: American Mathematical Soc.
ISBN: 1470429675
Category : Cluster algebras
Languages : en
Pages : 98
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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: Nima Arkani-Hamed
Publisher: Cambridge University Press
ISBN: 1316571645
Category : Science
Languages : en
Pages : 205
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Book Description
Outlining a revolutionary reformulation of the foundations of perturbative quantum field theory, this book is a self-contained and authoritative analysis of the application of this new formulation to the case of planar, maximally supersymmetric Yang–Mills theory. The book begins by deriving connections between scattering amplitudes and Grassmannian geometry from first principles before introducing novel physical and mathematical ideas in a systematic manner accessible to both physicists and mathematicians. The principle players in this process are on-shell functions which are closely related to certain sub-strata of Grassmannian manifolds called positroids - in terms of which the classification of on-shell functions and their relations becomes combinatorially manifest. This is an essential introduction to the geometry and combinatorics of the positroid stratification of the Grassmannian and an ideal text for advanced students and researchers working in the areas of field theory, high energy physics, and the broader fields of mathematical physics.
Author: Matteo Parisi
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
Author: David Jordan
Publisher: Cambridge University Press
ISBN: 1009103474
Category : Mathematics
Languages : en
Pages : 408
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Book Description
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: Ibrahim Assem
Publisher: Springer
ISBN: 3319745859
Category : Mathematics
Languages : en
Pages : 231
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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
