CLUSTER ALGEBRAS AND CLUSTER CATEGORIES ASSOCIATED WITH TRIANGULATED SURFACES: AN INTRODUCTION

For analytic mappings between Riemann surfaces a theorem of the Riesz type concerning so-called fine neighbourhood filters of Martin boundary points was proved by

We prove that if the zeroth cohomology of Γ is finite-dimensional, then the category per Γ/D b Γ is 2-CY and the image of the free dg module Γ is a cluster-tilting object

Let S be an oriented surface with punctures, and a finite number of special points on the boundary considered modulo isotopy. We prove that it is given by cluster

Using ([9], w 8.1.) it is possible to generalize the converse to Theorem 2 to the class of Riemann surfaces which are planar and have at least two but only a finite number