Cluster Algebras and Semipositive Symmetrizable Matrices

Cluster algebras are a class of commutative rings introduced by Fomin and Zelevinsky. It is well-known that these algebras are closely related with different areas of mathematics. A particular analogy exists between combinatorial aspects of cluster algebras and Kac-Moody algebras: roughly speaking, cluster algebras are associated with skew-symmetrizable matrices while Kac-Moody algebras correspond to symmetrizable generalized Cartan matrices. Both classes of algebras and the associated matrices have the same classification of finite type objects by the famous Cartan-Killing types. In this paper, we study an extension of this correspondence between the two classes of matrices to the affine type. In particular, we establish the cluster algebras which are determined by the generalized Cartan matrices of affine type. To state our results, we need some terminology. In this paper, we deal with the combinatorial aspects of the theory of cluster algebras, so we will not need their definition nor their algebraic properties. The main combinatorial objects of our study will be skew-symmetrizable matrices and the corresponding directed graphs. Let us recall that an integer matrix B is skew-symmetrizable if DB is skew-symmetric for some diagonal matrix D with positive diagonal entries. Recall also from [7] that, for any matrix index k, the mutation of a skew-symmetrizable matrix B in direction k is another skew-symmetrizable matrix μk(B) = B ′ whose entries are given as follows: B i,j = −Bi,j if i = k or j = k; otherwise B i,j = Bi,j + sgn(Bi,k)[Bi,kBk,j ]+ (where we use the notation [x]+ = max{x, 0} and sgn(x) = x/|x| with sgn(0) = 0). Mutation is an involutive operation, so repeated mutations in all directions give rise to the mutation-equivalence relation on skew-symmetrizable matrices. For each mutation (equivalence) class of skew-symmetrizable matrices, there is an associated cluster algebra [7]. In this paper, we will establish the mutation-classes which are naturally determined by the generalized Cartan matrices of affine type. We will use the following combinatorial construction: for a skew-symmetrizable n× n matrix B, its diagram is defined to be the directed graph Γ(B) whose vertices are the indices 1, 2, ..., n such that there is a directed edge from i to j if and only if Bij > 0, and this edge is assigned the weight |BijBji| . The diagram Γ(B) does not determine B as there could be several different skew-symmetrizable matrices whose diagrams are equal. In any case, we use the general term ”diagram” to mean the diagram of a skew-symmetrizable matrix. Then the mutation μk can be viewed as a transformation on diagrams (see Section 2 for a description) [7].