Seminar

The categorification of cluster algebras provides a powerful tool for understanding the subtle relationships between the elements of the cluster algebra. In the case of quantum cluster algebras, the categorification is only known for acyclic types.

In this talk we will restrict to the special case of rank two quantum cluster algebras (with two variables in each cluster and no coefficients).
I will develop the theory of quantum cluster algebras "from scratch", in particular giving a short elementary proof of the Laurent phenomenon.

I will introduce the notion of valued quivers and the basic features of their representation theory, namely reflection functors. I will show how one can construct non-initial cluster variables from the representations of a valued quiver, and if there is time, I will indicate how to prove this result using only the reflection functors.