Algebras of acyclic cluster type: tree type and type \(\widetilde A\). (Q2843351)

Let \(A\) be a finite dimensional algebra over an algebraically closed field, \(k\), of global dimension at most 2. Let \(D^b(A)\) denote its bounded derived category, with shift functor \(S\). By a result of \textit{C. Amiot} [Ann. Inst. Fourier 59, No. 6, 2525-2590 (2009; Zbl 1239.16011)], the orbit category \(D^b(A)/\tau^{-1}S\) can be embedded in a triangulated category \(C_A\), the \textit{generalized cluster category} of \(A\). This embedding is an equivalence if \(A\) is hereditary, and in this case the cluster category is as in [\textit{A. B. Buan} et al., Adv. Math. 204, No. 2, 572-618 (2006; Zbl 1127.16011)]. The algebra \(A\) can be regarded an object in \(C_A\), and its endomorphism ring, \(C(A)\), is known as a \textit{cluster-tilted algebra} if \(A\) is hereditary (see \textit{A. B. Buan} et al., [Trans. Am. Math. Soc. 359, No. 1, 323-332 (2007; Zbl 1123.16009)] and \textit{P. Caldero} et al., [Trans. Am. Math. Soc. 358, No. 3, 1347-1364 (2006; Zbl 1137.16020)]).NEWLINENEWLINE An algebra \(A\) of global dimension \(2\) is said to be of \textit{acyclic cluster type} if its generalized cluster category is equivalent to the cluster category of a hereditary algebra. It is said to be of \textit{cluster type} \(Q\) if its generalized cluster category is equivalent to the cluster category of the path algebra \(kQ\) of an acyclic quiver \(Q\). The main result is that if \(Q\) is an acyclic quiver whose underlying graph is a tree, and \(A\) is an algebra of global dimension \(2\) of cluster type \(Q\), then it is in fact derived equivalent to \(kQ\).NEWLINENEWLINE A description is given of the AR-quiver of the derived category of an algebra of cluster type affine type \(A\) (for any choice of orientation), and an explicit description of the algebras of affine type \(A\) cluster type is given, in terms of quivers with relations.