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By Michael Gekhtman

Cluster algebras, brought via Fomin and Zelevinsky in 2001, are commutative earrings with unit and no 0 divisors outfitted with a exclusive relations of turbines (cluster variables) grouped in overlapping subsets (clusters) of an identical cardinality (the rank of the cluster algebra) attached by means of trade family. Examples of cluster algebras comprise coordinate earrings of many algebraic kinds that play a popular function in illustration conception, invariant concept, the research of overall positivity, and so on. the speculation of cluster algebras has witnessed a wonderful development, firstly as a result of the many hyperlinks to a variety of matters together with illustration idea, discrete dynamical structures, Teichmüller conception, and commutative and non-commutative algebraic geometry. This publication is the 1st dedicated to cluster algebras. After featuring the required introductory fabric approximately Poisson geometry and Schubert types within the first chapters, the authors introduce cluster algebras and turn out their major homes in bankruptcy three. This bankruptcy should be considered as a primer at the conception of cluster algebras. within the last chapters, the emphasis is made on geometric elements of the cluster algebra conception, specifically on its family to Poisson geometry and to the speculation of integrable structures.

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