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Cluster algebras and triangulated surfaces. I: Cluster complexes. (English) Zbl 1263.13023
The authors initiate the study of an important class of cluster algebras which might be called cluster algebras from surfaces. The starting point for their construction is to consider triangulations of a connected oriented Riemann surface {S} with possibly empty boundary, and a non-empty finite set {M} of marked points on the closure of {S}, with at least one element of {M} in each connected component of boundaries. By introducing tagged arcs which are obtained by a tagging (plain or notched) for each end of an arc and modifying the concept of triangulations and the flipping procedure, the authors show that the adjacency matrix of a triangulation changes under flips that correspond to the mutation rule of the exchange matrix of the corresponding cluster algebra. Moreover, a natural bijection between tagged arcs (or tagged triangulations) and cluster variables (or clusters) is also given in the present paper. With the above setup, a series of conjectures of Fomin and Zelevinsky is verified for this class of cluster algebras from surfaces by exploiting their roots in combinatorial topology such as the intersection number of two arcs. For example, the seeds are determined by the clusters, and the exchange graph is independent of the choice of coefficients (see Theorem 5.6) and so on. In addition, the author show that cluster algebras of topological origin are mutation finite which by definition means the seeds appear only a finite number of exchanges. Moreover, the authors present briefly the \(11\) other known (exceptional) types of mutation finite cluster algebras with skew-symmetric exchange matrix. They wonder whether these are possibly all types of mutation finite skew symmetric cluster algebras or not. This has been recently confirmed in a work by Felikson, Shapiro and Tumarkin [“Skew–symmetric cluster algebras of finite mutation type”, arxiv:0811.1703].

13F60 Cluster algebras
57Q15 Triangulating manifolds
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
52B70 Polyhedral manifolds
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