Aslak Bakke Buan and Idun Reiten

Fomin and Zelevinsky [FZ1] have defined cluster algebras, and developed an interesting and influential theory about this class of algebras. We deal here with a special case of cluster algebras. Let B be an n×n integral skew symmetric matrix, or equivalently a finite quiver QB with n vertices and no loops or oriented cycles of length two. Let u = {u1, . . . , un} be a transcendence basis for F = Q(x1, . . . , xn). Then there is a cluster algebra associated to the pair (B, u). An essential ingredient in the definition of cluster algebras is the concept of mutation of matrices, or equivalently mutation of quivers. For each row i = 1, . . . , n in the skew-symmetric matrix B, there is an associated skew-symmetric matrix μi(B). If the quiver QB is a Dynkin quiver, it follows from [FZ1] that there is only a finite number of non-isomorphic quivers obtained from QB by sequences of mutations. The purpose of this paper is to show that whenQB is a connected quiver with no oriented cycles, then there is only a finite number of quivers obtained by sequences of mutations if and only if Q is Dynkin or extended Dynkin, or has at most two vertices. The first link from cluster algebras to tilting theory for finite dimensional algebras was discovered by Marsh, Reineke and Zelevinsky [MRZ]. This inspired the invention of cluster categories and cluster-tilted algebras [BMRRT, BMR1, BMR2]. Our main result is obtained as an application of this work. It answers a question by A. Seven, who has showed one implication (that finite mutation type implies Dynkin, extended Dynkin or rank two) using different methods [S1]. We would like to thank Otto Kerner for very helpful conversations.