SL_2-tilings and triangulations of the strip

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Abstract

SL_2-tilings were introduced by Assem, Reutenauer, and Smith in connection with frieses and their applications to cluster algebras. An SL_2-tiling is a bi-infinite matrix of positive integers such that each adjacent 2 x 2-submatrix has determinant 1. We construct a large class of new SL_2-tilings which contains the previously known ones. More precisely, we show that there is a bijection between our class of SL_2-tilings and certain combinatorial objects, namely triangulations of the strip.

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Cluster structures for 2-Calabi-Yau categories and unipotent groups

Aslak Buan, Idun Reiten, Jeanne Scott … (2007)

We investigate cluster tilting objects (and subcategories) in triangulated 2-Calabi-Yau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of non Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi-Yau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2-Calabi-Yau categories we construct cluster tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We give applications to cluster algebras and subcluster algebras related to unipotent groups, both in the Dynkin and non Dynkin case.