Generalized cluster algebras
Lead Research Organisation:
University of Leeds
Department Name: Pure Mathematics
Abstract
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.
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.
Organisations
Publications
Msapato D
(2021)
Counting the Number of t-Exceptional Sequences over Nakayama Algebras
in Algebras and Representation Theory
Msapato D
(2022)
Modular Fuss-Catalan numbers
in Discrete Mathematics
Msapato D
(2021)
The Karoubi envelope and weak idempotent completion of an extriangulated category
in Applied Categorical Structures
| Description | 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 have a connection to the representation theory of finite dimensional algebras, and they find applications in many things such as; quiver gauge theories in mathematical physics, and predicting the motion of shallow-water waves, in the work of Y. Kodama and L. Williams. of the motion Cluster algebras are defined in a certain way that allows us to understand some aspects of the representation theory of finite dimensional algebras. However, there is scope for generalizations to be found which could allow for a broader theory. One of the more studied cases of cluster algebras is known as the type A case. These are 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 that 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. The work of this research was to study aspects of representation theory. Of note, is the work in the publication "Modular Fuss-Catalan numbers". This work was motivated by wanting to understand a possible "higher-dimensional" generalisation of the combinatorics seen in the type A case. In particular, this work is concerned with the dissections of polygons known as m-angulations. These are dissections of polygons into polygons with m-sides, where m is an integer greater than or equal to 3. So when m is equal to three, these are precisely triangulations of polygons. This work specifically studied combinatorics which would provide the basis for the mutation rules that could be used to construct generalised cluster algebras of type A. Related to cluster algebras are objects known as cluster categories, which are examples of triangulated categories. Recently the notion of a triangulated category was generalized to that of an extriangulated category, in the work of H.Nakaoka and Y.Palu. Extriangulated categories thus provide a framework in which one can study cluster categories. In the paper, "The Karoubi envelope and weak idempotent completion of an extriangulated category", we sort to develop the theory of this framework which may aid in further understanding of cluster categories. In particular, we showed that the "idempotent completion" of an extriangulated category inherits an extriangulated structure. As a consequence, we provided new examples of extriangulated categories. The work in the publication " Counting the Number of t-Exceptional Sequences over Nakayama Algebras." was concerned with counting the representation theory objects known as t-exceptional sequences for a class of algebras known as the Nakayama algebras. The main results of this work were closed formulas expressing the number of t-exceptional sequences for certain Nakayama algebras and in some cases these established new connections to other areas of maths such as; the combinatorics of counting trees, and Lambert's W function which appears in statistics, fluid dynamics and many other areas of applied mathematics. |
| Exploitation Route | The work in the publication on "Modular Fuss-Catalan numbers" provides the basis for generalising the cluster algebras of type A to a higher dimensional in a natural way that recovers the known cluster algebras of type A. Consequently, this would then follow onto providing a good basis for a generalised cluster algebras for the other types. The work in the publication "The Karoubi envelope and weak idempotent completion of an extriangulated category" can be used as a blueprint for proving similar results for the categories known as n-exangulated categories. N-exangulated categories are a higher dimensional version of exangulated categories, which could provide a framework for studying cluster categories of the higher dimensional type A cluster algebras. Another consequence of our work is that it gives other research a methodology for constructing extriangulated categories with the property of being idempotent complete. This is a very desirable property in the theory of homology algebras with impactful consequences. The work in the publication "Counting the Number of t-Exceptional Sequences over Nakayama Algebras." revealed connections between the representation theory of Nakayama algebras and other areas of mathematics not known prior. The work also provided a method for counting t-Exceptional Sequences which can be applied to other classes of algebras and not just the Nakayama algebra. As a result, this work is a blueprint for work concerned with the combinatorics and counting of t-Exceptional Sequences. |
| Sectors | Other |