Tubular cluster algebras I: categorification

Abstract

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-Schröer using preprojective algebras.

Tubular cluster algebras II: Exponential growth

“…By the main result of [2], for a tubular cluster algebra the exchange graph of seeds is isomorphic to the exchange graph G of cluster tilting objects (in the corresponding tubular cluster category). Thus, by the considerations in Section 2.1, it is sufficient to construct a k-embedding of a tree T of exponential growth into the exchange graph G. This will be done in the next two sections by different methods.…”

“…Similarly as in [2] we define the complexity c(q) of q ∈ Q ∞ to be |d(q)| + r(q) + |d(q) − r(q)|. It follows easily that {1, 0, ∞} is the unique Farey-triple of minimal sum of the complexities, and that each other Farey triple can be mutated in a unique direction so that its sum of complexities decreases.…”

“…In Example 4.11 we have given a tilting sheaf realizing the Farey triple {0, 1, ∞} for the weight sequence (2, 2, 2, 2). We now consider the remaining weight sequences (3,3,3), (4, 4, 2) and (6,3,2). In each case we indicate how a tilting sheaf T realizing the Farey triple {0, 1, ∞} may be obtained from the canonical configuration T can by a sequence of mutations.…”

“…Tubular cluster algebras where introduced in [2] as a proper family of cluster algebras, due to their categorification by tubular cluster categories. These cluster algebras represent three of the 11 exceptional mutation finite cluster algebras with skew symmetric exchange matrix [8] and one is the surface algebra corresponding to the 4-punctured sphere.…”

“…Figure 1. Quivers associated to some elliptic root systems We refer to [2] for more details and context on tubular cluster algebras.…”