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Lecture Notes on Cluster Algebras
Robert J. Marsh, University of Leeds, United Kingdom
A publication of the European Mathematical Society.
cover
Zurich Lectures in Advanced Mathematics
2014; 122 pp; softcover
Volume: 19
ISBN-13: 978-3-03719-130-9
List Price: US$36
Member Price: US$28.80
Order Code: EMSZLEC/19
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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.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

Readership

Graduate students and research mathematicians interested in cluster algebras.

Table of Contents

  • Introduction
  • Cluster algebras
  • Exchange pattern cluster algebras
  • Reflection groups
  • Cluster algebras of finite type
  • Generalized associahedra
  • Periodicity
  • Quivers of finite mutation type
  • Grassmannians
  • Bibliography
  • Nomenclature
  • Index
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