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arXiv · 1705.10939

On indecomposable $\tau$-rigid modules over cluster-tilted algebras of tame type

Changjian Fu·Shengfei Geng
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Abstract

For a given cluster-tilted algebra $A$ of tame type, it is proved that different indecomposable $\tau$-rigid $A$-modules have different dimension vectors. This is motivated by Fomin-Zelevinsky's denominator conjecture for cluster algebras. As an application, we establish a weak version of the denominator conjecture for cluster algebras of tame type. Namely, we show that different cluster variables have different denominators with respect to a given cluster for a cluster algebra of tame type. Our approach involves Iyama-Yoshino's construction of subfactors of triangulated categories. In particular,we obtain a description of the subfactors of cluster categories of tame type with respect to an indecomposable rigid object, which is of independent interest.

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Changjian Fu, Shengfei Geng. 2017-05-31. On indecomposable $\tau$-rigid modules over cluster-tilted algebras of tame type. https://doi.org/10.1016/j.jalgebra.2019.04.024

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