Grassmannian Cluster Algebras and Quantum Groups

The project is set in pure mathematics in the areas of representation theory of associative algebras and Lie theory. Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. A representation makes an abstract algebraic object concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication. Representation theory is a powerful tool because it reduces problems in abstract algebra to problems in linear algebra, which is well understood. The algebraic objects that can be represented in such a way include groups, associative algebras and Lie algebras. One of the most important classes of associative algebras are cluster algebras. Cluster algebras are constructively defined commutative rings equipped with a distinguished set of generators (cluster variables) grouped into overlapping subsets (clusters) of the same finite cardinality (the rank of a cluster algebra). Among these algebras are coordinate rings of many algebraic varieties that play a prominent role in representation theory, invariant theory, the study of total positivity, etc. For instance, homogeneous coordinate rings of Grassmannians, Schubert varieties, and other related varieties carry a cluster algebra structure. Since its inception, the theory of cluster algebras has found a number of exciting connections and applications. Recent developments from Lie theory and higher representation theory opened up completely new perspectives. The inspiration for the current project comes from the connections of the representation theory of symmetric groups to the representation theory of Kac-Moody algebras, and from exciting new connections between canonical bases of quantum groups and cluster algebras that link many areas of mathematics. Our approach to the above families of algebras involves representation theoretical, combinatorial, homological, geometrical and computational methods. These methods represent a combination of classical methods, and new methods originating in Kac-Moody algebras, quantum groups and Grassmannian cluster algebras. Justification of such a choice of methods lies in the fact that this is one of the ground-breaking approaches that has the potential to produce very important results in a short time span. This approach is still in its development and it represents one of the most beautiful concepts in modern algebra. The goal of our project is to contribute to the structure theory of the algebras categorifying Grassmannian cluster algebras. In particular, we will study the structure of the maximal Cohen-Macaulay modules and graded structures on these algebras, as well as singularities of the quadratic forms associated to the affine Lie algebras arising from certain graphs and their connections to canonical bases of Grassmannian cluster algebras.

Grassmannian Cluster Algebras and Quantum Groups The project was set in pure mathematics in the areas of representation theory of associative algebras and Lie theory. Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. A representation makes an abstract algebraic object concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication. Representation theory is a powerful tool because it reduces problems in abstract algebra to problems in linear algebra, which is well understood. The algebraic objects that can be represented in such a way include groups, associative algebras and Lie algebras. One of the most important classes of associative algebras are cluster algebras. Cluster algebras are constructively defined commutative rings equipped with a distinguished set of generators (cluster variables) grouped into overlapping subsets (clusters) of the same finite cardinality (the rank of a cluster algebra). Among these algebras are coordinate rings of many algebraic varieties that play a prominent role in representation theory, invariant theory, the study of total positivity, etc. For instance, homogeneous coordinate rings of Grassmannians, Schubert varieties, and other related varieties carry a cluster algebra structure. Since its inception, the theory of cluster algebras has found a number of exciting connections and applications. Recent developments from Lie theory and higher representation theory opened up completely new perspectives. The inspiration for our project came from the connections of the representation theory of symmetric groups to the representation theory of Kac-Moody algebras, and from exciting new connections between canonical bases of quantum groups and cluster algebras that link many areas of mathematics. Our approach to the above families of algebras involved representation theoretical, combinatorial, homological, geometrical and computational methods. These methods represent a combination of classical methods, and new methods originating in Kac-Moody algebras, quantum groups and Grassmannian cluster algebras. Justification of such a choice of methods lies in the fact that this is one of the ground-breaking approaches that has the potential to produce very important results in a short time span. This approach is still in its development and it represents one of the most beautiful concepts in modern algebra. The goal of our project was to contribute to the structure theory of the algebras categorifying Grassmannian cluster algebras. In particular, we studied the structure of the maximal Cohen-Macaulay modules and graded structures on these algebras, singularities of the quadratic forms associated to the affine Lie algebras arising from certain graphs and their connections to canonical bases of Grassmannian cluster algebras. References: 1. K. Baur, D. Bogdanic, A. G. Elsener, Cluster Categories From Grassmannians and Root Combinatorics, Nagoya Mathematical Journal, 1-33, (2019).