Coloured quiver mutation for higher cluster categories.
\(m\)-cluster categories are \((m+1)\)-Calabi-Yau orbit categories of the derived category of a hereditary Abelian category. The authors give a combinatorial rule for the mutations in such a category. In a recent paper of \textit{O. Iyama} and \textit{Y. Yoshino} [Invent. Math. 172, No. 1, 117-168 (2008; Zbl 1140.18007)], it was shown that the existence of exchange triangles in cluster categories also holds for \(m\)-cluster categories. Furthermore, \textit{A. Wrålsen} [J. Algebra 321, No. 2, 532-547 (2009; Zbl 1177.18008)], and \textit{Y. Zhou, B. Zhu} [ J. Algebra 321, No. 10, 2898-2915 (2009; Zbl 1196.16013)] have proved that for an almost complete tilting object of an \(m\)-cluster category, there are \(m+1\) pairwise non-isomorphic complements given by \(m+1\) exchange triangles. The authors associate a quiver with coloured arrows of types \(\{0,\dots,m\}\) to each tilting object. For ordinary cluster categories (\(m=1\)), there is just the 0-coloured Gabriel quiver. Sequences of mutations can be performed thus in a purely combinatorial manner.
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