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Cluster Algebras and Poisson Geometry

Author : Michael Gekhtman,Michael Shapiro,Alek Vainshtein
Publisher : American Mathematical Soc.
Release : 2010
Category : Mathematics
ISBN : 9780821849729

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Book Cluster Algebras and Poisson Geometry Description/Summary:

Cluster algebras, introduced by Fomin and Zelevinsky in 2001, are commutative rings with unit and no zero divisors equipped with a distinguished family of generators (cluster variables) grouped in overlapping subsets (clusters) of the same cardinality (the rank of the cluster algebra) connected by exchange relations. Examples of cluster algebras include coordinate rings of many algebraic varieties that play a prominent role in representation theory, invariant theory, the study of total positivity, etc. The theory of cluster algebras has witnessed a spectacular growth, first and foremost due to the many links to a wide range of subjects including representation theory, discrete dynamical systems, Teichm�ller theory, and commutative and non-commutative algebraic geometry. This book is the first devoted to cluster algebras. After presenting the necessary introductory material about Poisson geometry and Schubert varieties in the first two chapters, the authors introduce cluster algebras and prove their main properties in Chapter 3. This chapter can be viewed as a primer on the theory of cluster algebras. In the remaining chapters, the emphasis is made on geometric aspects of the cluster algebra theory, in particular on its relations to Poisson geometry and to the theory of integrable systems.|Cluster algebras, introduced by Fomin and Zelevinsky in 2001, are commutative rings with unit and no zero divisors equipped with a distinguished family of generators (cluster variables) grouped in overlapping subsets (clusters) of the same cardinality (the rank of the cluster algebra) connected by exchange relations. Examples of cluster algebras include coordinate rings of many algebraic varieties that play a prominent role in representation theory, invariant theory, the study of total positivity, etc. The theory of cluster algebras has witnessed a spectacular growth, first and foremost due to the many links to a wide range of subjects including representation theory, discrete dynamical systems, Teichm�ller theory, and commutative and non-commutative algebraic geometry. This book is the first devoted to cluster algebras. After presenting the necessary introductory material about Poisson geometry and Schubert varieties in the first two chapters, the authors introduce cluster algebras and prove their main properties in Chapter 3. This chapter can be viewed as a primer on the theory of cluster algebras. In the remaining chapters, the emphasis is made on geometric aspects of the cluster algebra theory, in particular on its relations to Poisson geometry and to the theory of integrable systems.

Quantum Algebras and Poisson Geometry in Mathematical Physics

Author : Mikhail Vladimirovich Karasev,Elena M. Novikova,Yurii Mikhailovich Vorobjev
Publisher : American Mathematical Soc.
Release : 2005
Category : Mathematics
ISBN : 0821840401

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Book Quantum Algebras and Poisson Geometry in Mathematical Physics Description/Summary:

This collection presents new and interesting applications of Poisson geometry to some fundamental well-known problems in mathematical physics. The methods used by the authors include, in addition to advanced Poisson geometry, unexpected algebras with non-Lie commutation relations, nontrivial (quantum) Kahlerian structures of hypergeometric type, dynamical systems theory, semiclassical asymptotics, etc.

Lectures on Poisson Geometry

Author : Marius Crainic,Rui Loja Fernandes,Ioan Mărcuţ
Publisher : American Mathematical Soc.
Release : 2021-10-14
Category : Education
ISBN : 9781470466671

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Book Lectures on Poisson Geometry Description/Summary:

This excellent book will be very useful for students and researchers wishing to learn the basics of Poisson geometry, as well as for those who know something about the subject but wish to update and deepen their knowledge. The authors' philosophy that Poisson geometry is an amalgam of foliation theory, symplectic geometry, and Lie theory enables them to organize the book in a very coherent way. —Alan Weinstein, University of California at Berkeley This well-written book is an excellent starting point for students and researchers who want to learn about the basics of Poisson geometry. The topics covered are fundamental to the theory and avoid any drift into specialized questions; they are illustrated through a large collection of instructive and interesting exercises. The book is ideal as a graduate textbook on the subject, but also for self-study. —Eckhard Meinrenken, University of Toronto

Algebraic Geometry and Number Theory

Author : victor ginzburg
Publisher : Springer Science & Business Media
Release : 2007-12-31
Category : Mathematics
ISBN : 9780817645328

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Book Algebraic Geometry and Number Theory Description/Summary:

This book represents a collection of invited papers by outstanding mathematicians in algebra, algebraic geometry, and number theory dedicated to Vladimir Drinfeld. Original research articles reflect the range of Drinfeld's work, and his profound contributions to the Langlands program, quantum groups, and mathematical physics are paid particular attention. These ten original articles by prominent mathematicians, dedicated to Drinfeld on the occasion of his 50th birthday, broadly reflect the range of Drinfeld's own interests in algebra, algebraic geometry, and number theory.

Symmetries and Integrability of Difference Equations

Author : Decio Levi,Raphaël Rebelo,Pavel Winternitz
Publisher : Springer
Release : 2017-06-30
Category : Science
ISBN : 9783319566665

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Book Symmetries and Integrability of Difference Equations Description/Summary:

This book shows how Lie group and integrability techniques, originally developed for differential equations, have been adapted to the case of difference equations. Difference equations are playing an increasingly important role in the natural sciences. Indeed, many phenomena are inherently discrete and thus naturally described by difference equations. More fundamentally, in subatomic physics, space-time may actually be discrete. Differential equations would then just be approximations of more basic discrete ones. Moreover, when using differential equations to analyze continuous processes, it is often necessary to resort to numerical methods. This always involves a discretization of the differential equations involved, thus replacing them by difference ones. Each of the nine peer-reviewed chapters in this volume serves as a self-contained treatment of a topic, containing introductory material as well as the latest research results and exercises. Each chapter is presented by one or more early career researchers in the specific field of their expertise and, in turn, written for early career researchers. As a survey of the current state of the art, this book will serve as a valuable reference and is particularly well suited as an introduction to the field of symmetries and integrability of difference equations. Therefore, the book will be welcomed by advanced undergraduate and graduate students as well as by more advanced researchers.

Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths

Author : Sergey Fomin,Professor Dylan Thurston
Publisher : American Mathematical Soc.
Release : 2018-10-03
Category : Cluster algebras
ISBN : 9781470429676

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Book Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths Description/Summary:

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.

Quantum Cluster Algebra Structures on Quantum Nilpotent Algebras

Author : K. R. Goodearl,M. T. Yakimov
Publisher : American Mathematical Soc.
Release : 2017-04-25
Category : Algebra
ISBN : 9781470436940

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Book Quantum Cluster Algebra Structures on Quantum Nilpotent Algebras Description/Summary:

All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it is shown that these quantum cluster algebras always equal the corresponding upper quantum cluster algebras. Previous approaches to these problems for the construction of (quantum) cluster algebra structures on (quantized) coordinate rings arising in Lie theory were done on a case by case basis relying on the combinatorics of each concrete family. The results of the paper have a broad range of applications to these problems, including the construction of quantum cluster algebra structures on quantum unipotent groups and quantum double Bruhat cells (the Berenstein-Zelevinsky conjecture), and treat these problems from a unified perspective. All such applications also establish equality between the constructed quantum cluster algebras and their upper counterparts

Exotic Cluster Structures on $SL_n$: The Cremmer-Gervais Case

Author : M. Gekhtman,M. Shapiro,A. Vainshtein
Publisher : American Mathematical Soc.
Release : 2017-02-20
Category : Cluster algebras
ISBN : 9781470422585

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Book Exotic Cluster Structures on $SL_n$: The Cremmer-Gervais Case Description/Summary:

This is the second paper in the series of papers dedicated to the study of natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson–Lie structures compatible with these cluster structures. According to our main conjecture, each class in the Belavin–Drinfeld classification of Poisson–Lie structures on corresponds to a cluster structure in . The authors have shown before that this conjecture holds for any in the case of the standard Poisson–Lie structure and for all Belavin–Drinfeld classes in , . In this paper the authors establish it for the Cremmer–Gervais Poisson–Lie structure on , which is the least similar to the standard one.

Algebra, Geometry, and Physics in the 21st Century

Author : Denis Auroux,Ludmil Katzarkov,Tony Pantev,Yan Soibelman,Yuri Tschinkel
Publisher : Birkhäuser
Release : 2017-07-27
Category : Mathematics
ISBN : 9783319599397

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Book Algebra, Geometry, and Physics in the 21st Century Description/Summary:

This volume is a tribute to Maxim Kontsevich, one of the most original and influential mathematicians of our time. Maxim’s vision has inspired major developments in many areas of mathematics, ranging all the way from probability theory to motives over finite fields, and has brought forth a paradigm shift at the interface of modern geometry and mathematical physics. Many of his papers have opened completely new directions of research and led to the solutions of many classical problems. This book collects papers by leading experts currently engaged in research on topics close to Maxim’s heart. Contributors: S. Donaldson A. Goncharov D. Kaledin M. Kapranov A. Kapustin L. Katzarkov A. Noll P. Pandit S. Pimenov J. Ren P. Seidel C. Simpson Y. Soibelman R. Thorngren

Higher Structures in Geometry and Physics

Author : Alberto S. Cattaneo,Anthony Giaquinto,Ping Xu
Publisher : Springer Science & Business Media
Release : 2010-11-25
Category : Mathematics
ISBN : 9780817647353

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Book Higher Structures in Geometry and Physics Description/Summary:

This book is centered around higher algebraic structures stemming from the work of Murray Gerstenhaber and Jim Stasheff that are now ubiquitous in various areas of mathematics— such as algebra, algebraic topology, differential geometry, algebraic geometry, mathematical physics— and in theoretical physics such as quantum field theory and string theory. These higher algebraic structures provide a common language essential in the study of deformation quantization, theory of algebroids and groupoids, symplectic field theory, and much more. Each contribution in this volume expands on the ideas of Gerstenhaber and Stasheff. The volume is intended for post-graduate students, mathematical and theoretical physicists, and mathematicians interested in higher structures.

Log-canonical Poisson Structures and Non-commutative Integrable Systems

Author : Nicholas Ovenhouse
Publisher : Unknown
Release : 2019
Category : Electronic dissertations
ISBN : 139219041X

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Book Log-canonical Poisson Structures and Non-commutative Integrable Systems Description/Summary:

Log-canonical Poisson structures are a particularly simple type of bracket which are given by quadratic expressions in local coordinates. They appear in many places, including the study of cluster algebras. A Poisson bracket is "compatible" with a cluster algebra structure if the bracket is log-canonical with respect to each cluster. In joint work with John Machacek, we prove a structural result about such Poisson structures, which justifies the use and significance of such brackets in cluster theory. The result says that no rational coordinate-changes can transform these brackets into a simpler form. The pentagram map is a discrete dynamical system on the space of plane polygons first intro- duced by Schwartz in 1992. It was proved to be Liouville integrable by Schwartz, Ovsienko, and Tabachnikov in 2010. Gekhtman, Shapiro, and Vainshtein studied Poisson geometry associated to certain networks embedded in a disc or annulus, and its relation to cluster algebras. These Poisson structures are log-canonical. Later, Gekhtman et al. and Tabachnikov reinterpreted the pentagram map in terms of these networks, and used the associated Poisson structures to give a new proof of integrability. In 2011, Mari Beffa and Felipe introduced a generalization of the pentagram map to certain Grassmannians, and proved it had a Lax representation. We reinterpret this Grassmann penta- gram map in terms of non-commutative algebra, in particular the double brackets of Van den Bergh, and generalize the approach of Gekhtman et al. to establish a non-commutative version of integrability. The integrability of the pentagram maps in both projective space and the Grass-mannian follow from this more general algebraic system by projecting to representation spaces.

Developments and Retrospectives in Lie Theory

Author : Geoffrey Mason,Ivan Penkov,Joseph A. Wolf
Publisher : Springer
Release : 2014-11-12
Category : Mathematics
ISBN : 9783319099347

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Book Developments and Retrospectives in Lie Theory Description/Summary:

The Lie Theory Workshop, founded by Joe Wolf (UC, Berkeley), has been running for over two decades. At the beginning, the top universities in California and Utah hosted the meetings, which continue to run on a quarterly basis. Experts in representation theory/Lie theory from various parts of the US, Europe, Asia (China, Japan, Singapore, Russia), Canada, and South and Central America were routinely invited to give talks at these meetings. Nowadays, the workshops are also hosted at universities in Louisiana, Virginia, and Oklahoma. These Lie theory workshops have been sponsored by the NSF, noting the talks have been seminal in describing new perspectives in the field covering broad areas of current research. The contributors have all participated in these Lie theory workshops and include in this volume expository articles which will cover representation theory from the algebraic, geometric, analytic, and topological perspectives with also important connections to math physics. These survey articles, review and update the prominent seminal series of workshops in representation/Lie theory mentioned above, and reflects the widespread influence of those workshops in such areas as harmonic analysis, representation theory, differential geometry, algebraic geometry, number theory, and mathematical physics. Many of the contributors have had prominent roles in both the classical and modern developments of Lie theory and its applications.

Lie Groups, Geometry, and Representation Theory

Author : Victor G. Kac,Vladimir L. Popov
Publisher : Springer
Release : 2018-12-12
Category : Mathematics
ISBN : 9783030021917

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Book Lie Groups, Geometry, and Representation Theory Description/Summary:

This volume, dedicated to the memory of the great American mathematician Bertram Kostant (May 24, 1928 – February 2, 2017), is a collection of 19 invited papers by leading mathematicians working in Lie theory, representation theory, algebra, geometry, and mathematical physics. Kostant’s fundamental work in all of these areas has provided deep new insights and connections, and has created new fields of research. This volume features the only published articles of important recent results of the contributors with full details of their proofs. Key topics include: Poisson structures and potentials (A. Alekseev, A. Berenstein, B. Hoffman) Vertex algebras (T. Arakawa, K. Kawasetsu) Modular irreducible representations of semisimple Lie algebras (R. Bezrukavnikov, I. Losev) Asymptotic Hecke algebras (A. Braverman, D. Kazhdan) Tensor categories and quantum groups (A. Davydov, P. Etingof, D. Nikshych) Nil-Hecke algebras and Whittaker D-modules (V. Ginzburg) Toeplitz operators (V. Guillemin, A. Uribe, Z. Wang) Kashiwara crystals (A. Joseph) Characters of highest weight modules (V. Kac, M. Wakimoto) Alcove polytopes (T. Lam, A. Postnikov) Representation theory of quantized Gieseker varieties (I. Losev) Generalized Bruhat cells and integrable systems (J.-H. Liu, Y. Mi) Almost characters (G. Lusztig) Verlinde formulas (E. Meinrenken) Dirac operator and equivariant index (P.-É. Paradan, M. Vergne) Modality of representations and geometry of θ-groups (V. L. Popov) Distributions on homogeneous spaces (N. Ressayre) Reduction of orthogonal representations (J.-P. Serre)

Proceedings Of The International Congress Of Mathematicians 2010 (Icm 2010) (In 4 Volumes) - Vol. I: Plenary Lectures And Ceremonies, Vols. Ii-iv: Invited Lectures

Author : Bhatia Rajendra,Pal Arup,Rangarajan G
Publisher : World Scientific
Release : 2011-06-06
Category : Mathematics
ISBN : 9789814462938

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Book Proceedings Of The International Congress Of Mathematicians 2010 (Icm 2010) (In 4 Volumes) - Vol. I: Plenary Lectures And Ceremonies, Vols. Ii-iv: Invited Lectures Description/Summary:

ICM 2010 proceedings comprises a four-volume set containing articles based on plenary lectures and invited section lectures, the Abel and Noether lectures, as well as contributions based on lectures delivered by the recipients of the Fields Medal, the Nevanlinna, and Chern Prizes. The first volume will also contain the speeches at the opening and closing ceremonies and other highlights of the Congress.

Lecture Notes on Cluster Algebras

Author : Robert J. Marsh
Publisher : Erich Schmidt Verlag GmbH & Co. KG
Release : 2013
Category : Mathematics
ISBN : 3037191309

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Book Lecture Notes on Cluster Algebras Description/Summary:

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.

Nonlinear Systems and Their Remarkable Mathematical Structures

Author : Norbert Euler,Maria Clara Nucci
Publisher : CRC Press
Release : 2019-12-06
Category : Mathematics
ISBN : 9780429554308

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Book Nonlinear Systems and Their Remarkable Mathematical Structures Description/Summary:

Nonlinear Systems and Their Remarkable Mathematical Structures, Volume 2 is written in a careful pedagogical manner by experts from the field of nonlinear differential equations and nonlinear dynamical systems (both continuous and discrete). This book aims to clearly illustrate the mathematical theories of nonlinear systems and its progress to both non-experts and active researchers in this area. Just like the first volume, this book is suitable for graduate students in mathematics, applied mathematics and engineering sciences, as well as for researchers in the subject of differential equations and dynamical systems. Features Collects contributions on recent advances in the subject of nonlinear systems Aims to make the advanced mathematical methods accessible to the non-experts Suitable for a broad readership including researchers and graduate students in mathematics and applied mathematics

Proceedings Of The International Congress Of Mathematicians 2018 (Icm 2018) (In 4 Volumes)

Author : Sirakov Boyan,Souza Paulo Ney De,Viana Marcelo
Publisher : World Scientific
Release : 2019-02-27
Category : Mathematics
ISBN : 9789813272897

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Book Proceedings Of The International Congress Of Mathematicians 2018 (Icm 2018) (In 4 Volumes) Description/Summary:

The Proceedings of the ICM publishes the talks, by invited speakers, at the conference organized by the International Mathematical Union every 4 years. It covers several areas of Mathematics and it includes the Fields Medal and Nevanlinna, Gauss and Leelavati Prizes and the Chern Medal laudatios.

Triangulated Categories

Author : Thorsten Holm,Peter Jørgensen,Raphaël Rouquier
Publisher : Cambridge University Press
Release : 2010-06-24
Category : Mathematics
ISBN : 9781139488884

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Book Triangulated Categories Description/Summary:

Over the last few decades triangulated categories have become increasingly important, to the extent that they can now be viewed as a unifying theory underlying major parts of modern mathematics. This 2010 collection of survey articles, written by leading experts, covers fundamental aspects of triangulated categories, as well as applications in algebraic geometry, representation theory, commutative algebra, microlocal analysis and algebraic topology. These self-contained articles are a useful introduction for graduate students entering the field and a valuable reference for experts.

Proceedings of the International Congress of Mathematicians

Author : Rajendra Bhatia
Publisher : World Scientific
Release : 2011-06-06
Category : Mathematics
ISBN : 9789814324359

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Book Proceedings of the International Congress of Mathematicians Description/Summary:

ICM 2010 proceedings comprise a four-volume set containing articles based on plenary lectures and invited section lectures, the Abel and Noether lectures, as well as contributions based on lectures delivered by the recipients of the Fields Medal, the Nevanlinna, and Chern Prizes. The first volume will also contain the speeches at the opening and closing ceremonies and other highlights of the Congress