By Michael Gekhtman
Cluster algebras, brought by means of Fomin and Zelevinsky in 2001, are commutative earrings with unit and no 0 divisors built with a unusual relatives of turbines (cluster variables) grouped in overlapping subsets (clusters) of an analogous cardinality (the rank of the cluster algebra) hooked up via alternate family. Examples of cluster algebras comprise coordinate earrings of many algebraic types that play a well-known position in illustration thought, invariant conception, the learn of overall positivity, and so forth. the idea of cluster algebras has witnessed a staggering progress, before everything because of the many hyperlinks to a variety of topics together with illustration idea, discrete dynamical platforms, TeichmÃ¼ller thought, and commutative and non-commutative algebraic geometry. This publication is the 1st dedicated to cluster algebras. After providing the mandatory introductory fabric approximately Poisson geometry and Schubert kinds within the first chapters, the authors introduce cluster algebras and end up their major houses in bankruptcy three. This bankruptcy will be seen as a primer at the thought of cluster algebras. within the ultimate chapters, the emphasis is made on geometric elements of the cluster algebra conception, specifically on its kin to Poisson geometry and to the idea of integrable platforms.
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