Algebraic Geometry Seminar: Dylan Rupel (Oregon)

A quantum cluster algebra is a subalgebra of an ambient skew field of rational functions in finitely many indeterminates. The quantum cluster algebra is generated by a (usually infinite) recursively defined collection called the cluster variables. Explicit expressions for the cluster variables are difficult to compute on their own as the recursion describing them involves division inside this skew field. In this talk I will describe the rank 2 cluster variables explicitly by relating them to varieties associated to valued representations of a quiver with 2 vertices. I will also indicate to what extent the theory. I present is applicable to higher rank quantum cluster algebras.