Tubular cluster algebras I: categorification

Understand

We present a categorification of four mutation finite cluster algebras by the cluster category of the category of coherent sheaves over a weighted projective line of tubular weight type.

• Each of these cluster algebras which we call tubular is associated to an elliptic root system.
• We show that via a cluster character the cluster variables are in bijection with the positive real Schur roots associated to the weighted projective line.
• In one of the four cases this is achieved by the approach to cluster algebras of Fomin-Shapiro-Thurston using a 2-sphere with 4 marked points whereas in the remaining cases it is done by the approach of Geiss-Leclerc-Schroer using preprojective algebras.