Generalized cluster algebras

环是一个包含元素的数学对象，这些元素可以以两种不同的方式组合：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确立，但更复杂的环需要更详细的描述。环可以用诸如矩阵（数字数组）之类的对象来描述，或者通过给出基本生成器并指定它们之间的关系来描述。环的元素则是服从这些关系的生成元的形式积。簇代数是由Sergey Fomin和Andrei Zelevinsky在2001年引入的环，它使用一种革命性的方法来定义，在这种方法中，只有一个小的初始集合，或者说是生成子的簇。然后，这些初始发电机“突变”形成新的集群，最终形成整个发电机组。这些关系是由产生器的形式自然产生的。簇代数是为了解决李论中的一个关键问题而引入的，但它们在表示理论中有很强的应用，在表示理论中，抽象的数学对象是通过用更实际的对象代替它们来研究的。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。具有有限多个生成器的聚类代数由Fomin和Zelevinsky分类，并以某些图形（称为Dynkin图）进行了漂亮的描述。其中研究最多的是A型代数，它包含簇代数，其生成器对应于多边形中连接对顶点的对角线。集群是指定的生成器集合，对应于多边形的三角剖分——将多边形划分为三角形的对角线集合，或者，也可以是多边形不相交的对角线的最大集合。在A型案例中，与表征理论的关系非常强烈和明确。然而，聚类代数只能用于描述表征理论的有限方面。它们是以一种非常特殊的方式定义的，这可能会使其一般化。这个项目的目的是寻找新的、更一般的簇代数。预计该项目将创建相关的表示理论，并沿着与已知案例类似的路线发展。需要开发的新方法包括定义与（广义）聚类代数对应的表示的新方法、新类型的突变规则以及从（广义）聚类代数产生代数对象的新方法。聚类代数和表示理论之间的联系是国际上许多研究者十多年来发展起来的，取得了丰硕的成果，并在聚类代数和表示理论中得到了重要的应用。因此，我们有理由期待一个更普遍的理论在数学中也有很好的应用。簇代数也出现在数学物理中（例如，在颤振规范理论中），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，有关簇代数的思想也被用于预测浅水波浪的运动，并且也有可能与该理论发展关系。这个项目的工作主要是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.