On the Cluster Multiplication Theorem for Acyclic Cluster Algebras

In [3] and [13], the authors proved the cluster multiplication theorems for finite type and affine type. We generalize their results and prove the cluster multiplication theorem for arbitrary type by using the properties of 2–Calabi–Yau (Auslander–Reiten formula) and high order associativity. Introduction Cluster algebras were introduced by Fomin and Zelevinsky in [9]. By definition, the cluster algebras are commutative algebras generated by a set of variables called cluster variables. Let Q be a quiver and we denote by A(Q) the associated cluster algebra. If Q does not contain oriented cycles, we call Q an acyclic quiver. The cluster algebras associated to acyclic quivers are called acyclic cluster algebras. Their relations to quiver representations were first revealed in [19]. In [1], the authors found the general framework of the link of cluster algebras and quiver representations and introduced the cluster categories as the categorification of acyclic cluster algebras. For an acyclic quiver Q, the associated cluster category C(Q) is the orbit category of the bounded derived category D(modkQ) over a field k by the auto–equivalence F := [1]τ where [1] is the translation functor and τ is the AR-translation. In general, one can define the cluster category of a hereditary category with Serre duality ν by taking τ = [−1]ν as shown in [16]. In [2], the authors introduced a certain structure of Hall algebra involving the cluster category C(Q) by associating the objects in C(Q) to some variables given by an explicit map X?, called the Caldero-Chapoton map. We denote by XM the variable (called the generalized cluster variable) associated to an object M in C(Q). In the case that M is a non-projective kQ-module, the authors gave the the multiplication of XM and XτM as follows: (0.1) XτMXM = XB + 1 where B is the middle term of the almost split sequence involving M and τM. If Q is a simply laced Dynkin quiver, Caldero and Keller [3] extended the above multiplication (0.1) to the multiplication of any two variables associated to two indecomposable objects in C(Q) as follows χc(PExt (M,N))XMXN = ∑ Y (χc(PExt (M,N)Y ) + χc(PExt (N,M)Y ))XY Date: November 15, 2007. 2000 Mathematics Subject Classification. Primary 16G20, 16G70; Secondary 14M99, 18E30.