DescriptionCluster algebras have been introduced a decade ago by Sergey Fomin and Andrei Zelevinsky in relation to problems in combinatorics and Lie groups. Since then, a number of exciting connections and applications to various branches of mathematics and mathematical physics have been found (e.g. representation theory, Poisson geometry, Teichmueller theory, and many others).In the simplest, coefficient-free case, a cluster algebra is a subring of the field of rational functions in a finite number of variables. The algebra is defined using initial data, namely a skew-symmetric matrix (called exchange matrix) and a free generating set of the field (called initial cluster), known as an initial seed. Exchange matrices and clusters undergo some combinatorial procedure (known as mutation), iterated applications of which give rise to a (usually infinite) collection of rational functions (called cluster variables); this is then a generating set for the cluster algebra. Cluster algebras carry an interesting combinatorial structure. For example, those whose set of cluster variables is finite are classified by the Dynkin diagrams. The cluster algebra corresponding to a Dynkin diagram is closely related to the corresponding root system. In some more general cases relations to infinite root systems can be established. Prerequisites and suggestionsFor this rather demanding topic a very good background in algebra is needed. Unfortunately there is no textbook on the subject yet, so one will have to read the original sources.We can suggest that the student treat the following questions:
ResourcesThe current state of the developments related to the field, including links to papers, seminars, etc., is represented at the online Cluster Algebra Portal created and maintained by Sergey Fomin. A selection of somewhat introductory material can be found here:A. Zelevinsky, What is a cluster Algebra? S. Fomin, N. Reading, Root systems and generalized associahedra, notes from summer school S. Fomin, A. Zelevinsky, Cluster Algebras I: Foundations S. Fomin, A. Zelevinsky, Cluster Algebras II: Finite Type Classification M. Gekhtman, M. Shapiro, and A. Vainshtein, Cluster algebras and Poisson geometry S. Fomin, M. Shapiro, and D. Thurston, Cluster algebras and triangulated surfaces. Part I: Cluster complexes J. E. Humphreys, Reflection groups and Coxeter groups Library link |
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