# Lecture Notes on Cluster Algebras

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*Robert J. Marsh*

A publication of the European Mathematical Society

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.