Generalized cluster algebras

环是一个包含元素的数学对象，这些元素可以通过两种不同的方式组合在一起：相加或相乘。一个关键的例子是整数集。这很容易描述，因为整数及其性质已经很好地建立了，但更复杂的环需要更详细的描述。环可以用诸如矩阵(数组)之类的对象来描述，或者通过给出基本生成元并指定它们之间的关系来描述。簇代数是由Sergey Fomin和Andrei Zlevinsky于2001年提出的环，它用一种革命性的方法定义了环，在这种方法中，只指定了一小部分初始生成元。然后，这些初始的发电机被“突变”，形成新的集群，最终形成整个发电机组。这种关系是由生成元的形式自然产生的。簇代数最初是为了解决李论中的一个关键问题而引入的，但它们在表示论中得到了很强的应用。在表示论中，抽象的数学对象是通过用更多实际的对象来研究的，例如以与原始元素相同的方式进行加法和乘法的数字或矩阵。具有有限多个生成元的簇代数由Fomin和Zlevinsky分类，并用某些图进行了漂亮的描述，称为动态图。其中研究最多的是A型情形，它包含生成元对应于连接顶点对的多边形中的对角线的簇代数。作为生成器的指定集合的簇对应于多边形的三角剖分-将多边形分成三角形的对角线集合，或者可替换地，不相交的多边形的对角线的最大集合。在A类情况下，它与表示理论的关系是非常强和清晰的，但是，簇代数只能用来描述表示理论的有限方面。它们是以一种非常特殊的方式定义的，这可能允许泛化。这个项目的目的将是寻找新的、更通用的簇代数类型。预计该项目将创造相关的表征理论，并沿着与已知案例类似的路线发展。新的方法包括定义对应于(广义)簇代数的表示的新方法，新类型的变异规则，以及从(广义)簇代数产生代数对象的新方法。簇代数与表示理论之间的联系由国际上许多研究人员在10多年的时间里发展起来，取得了非常丰硕的成果，并导致了在簇代数和表示理论方面的重要应用。因此，我们有理由期待一个更一般的理论在数学中也会有很好的应用。团簇代数也出现在数学物理中(例如，在箭图规范理论中)，因此这也将是一个可能有潜在应用的领域。在Y.KoDama和L.Williams的工作中，与簇代数有关的思想也被用来预测浅水波的运动，并有可能与这一理论建立联系。这个项目的工作主要集中在代数和组合学领域。

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.