Categorification of cluster algebras

Quivers are a very broad class of objects that were originally intended to study subspaces of a given vector space. On the route to classifying all the semisimple Lie algebras it was discovered that certain diagrams play a very important role in Lie theory (the so-called Dynkin diagrams). If we add an orientation to such a diagram we obtain a Dynkin quiver.


Thanks to work by Gabriel and Ringel it turns out that quiver representation theory over a finite field can categorify the positive part of the quantum envelopping algebra of the same type. In his incredible paper Lusztig managed to construct a new basis of the quantum envelopping algebra which also gives bases of all of the highest weight irreducible representations.


Motivated by ideas related to canonical bases Fomin and Zelevinsky constructed an algebra by starting from a finite set and then "mutating". The algebras generated by the mutations are cluster algebras. This mutation process can be related to a mutation of quivers.


In a paper by Buan, Marsh, Reineke, Reiten and Todorov a certain quotient of the bound derived category of modules over the path algebra of a related quiver was shown to give us the categorical structure of a cluster algebra in the case of acyclic type.


The aim of this project would be to search for a categorification of cluster algebras in a different case (the geometric case coming from grassmannians).

This sort of work could hope to find applications in algebra, geometry, combinatorics and even mathematical physics.