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Lecture Notes on Cluster Algebras
Zurich Lectures in Advanced Mathematics

Robert J. Marsh (University of Leeds, UK)

Lecture Notes on Cluster Algebras

ISBN print 978-3-03719-130-9, ISBN online 978-3-03719-630-4
DOI 10.4171/130
January 2014, 121 pages, softcover, 17 x 24 cm.
28.00 Euro

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. The 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.

Keywords: Associahedron, cluster algebra, cluster complex, Dynkin diagram, finite mutation type, Grassmannian, Laurent phenomenon, reflection group, periodicity, polytope, quiver mutation, root system, Somos sequence, surface

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