Rigid objects in higher cluster categories. (English) Zbl 1177.18008
This is a study about the maximal \(m\)-rigid objects and \(m\)-cluster tilting objects in the \(m\)-cluster category \(\mathcal{C}_H^m\) of a finite-dimensional hereditary algebra \(H\) with \(n\) nonisomorphic simple modules. The results of this paper are, first, any maximal \(m\)-rigid object in \(\mathcal{C}_H^m\) has \(n\) indecomposable summands, where \(n\) is the number of nonisomorphic simple modules in \(mod H.\) Second, any \(m\)-rigid object having \(n-1\) indecomposable nonisomorphic summands has \(m+1\) indecomposable complements. The author also shows that all maximal \(m\)-rigid objects in \(\mathcal{C}_H^m\) coincide with \(m\)-cluster tilting objects; and if \(\Gamma = \text{End}_{\mathcal{C}_H}(T)^{\text{op}}\) where \(T\) is a maximal rigid object in \(\mathcal{C}_H,\) i.e., \(\Gamma\) is a cluster-tilted algebra, then so is \(\Gamma/e \Gamma,\) where \(e\) is an idempotent associated with a vertex of \(\Gamma.\) The main idea of this paper is to use the techniques by Buan, Marsh and Reiten and generalized version of these to prove most of the main results by induction.
Reviewer: Intan Muchtadi-Alamsyah (Bandung)
MSC:
| 18E30 | Derived categories, triangulated categories (MSC2010) |
| 16G20 | Representations of quivers and partially ordered sets |
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