Generalized cluster algebras

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A ring is a mathematical object containing elements which can be combined in two different ways: either added together or multiplied. A key example is the set of whole numbers. This is easy to describe since whole numbers and their properties are well-established, but more complex rings need more detailed descriptions. Rings might be described in terms of objects such as matrices (arrays of numbers) or by giving fundamental generators and specifying the relationship between them. The elements of the ring are then formal products of the generators subject to the relations.Cluster algebras are rings that were introduced in 2001 by Sergey Fomin and Andrei Zelevinsky, defined using a revolutionary method in which only a small initial set, or cluster, of generators is specified. These initial generators are then 'mutated' to form new clusters, eventually giving the entire generating set. The relations arise naturally from the form of the generators.Cluster algebras were introduced as an attempt to solve a key problem in Lie theory, but they turned out to have strong applications to representation theory, where abstract mathematical objects are studied by replacing them with more down-to-earth objects., such as numbers or matrices which add and multiply in the same way as the original elements.The cluster algebras with finitely many generators were classified by Fomin and Zelevinsky, and have a beautiful description in terms of certain graphs, known as Dynkin diagrams. The most studied of these, known as the type A case, contains cluster algebras whose generators correspond to diagonals in a polygon joining pairs of vertices. The clusters, which are specified collections of generators, correspond to triangulations of a polygon - collections of diagonals which divide the polygon into triangles or, alternatively, maximal collections of diagonals of the polygon which do not cross. In the type A case, the relationship with representation theory is very strong and clear.However, cluster algebras can only be used to describe limited aspects of representation theory. They are defined in a very particular way which may allow for generalization. The aim of this project will be to find new, more general kinds of cluster algebras. It is also expected that the project will create related representation theory, and develop it along lines analogous to that in the known cases. Novel methods to be developed include new ways to define representations corresponding to (generalized) cluster algebras, new kinds of mutation rules, and new ways of producing algebraic objects from (generalized) cluster algebras.The connection between cluster algebras and representation theory, developed by many researchers internationally over more than 10 years, has been very fruitful and has led to important applications both to cluster algebras and to representation theory. So it is reasonable to expect that a more general theory will also have good applications in mathematics. Cluster algebras also appear in mathematical physics (for example, in quiver gauge theories) and so this will also be an area where there might be potential applications. Ideas relating to cluster algebras have also been used to predict the motion of shallow water waves, in work of Y. Kodama and L. Williams, and there is the potential for a relationship to be developed with this theory also.The work of this project lies primarily in the areas of algebra and combinatorics.