Cluster Algebras I : Foundations

In this paper, we initiate the study of a new class of algebras, which we call cluster algebras. Before giving precise definitions, we present some of the main features of these algebras. For any positive integer n, a cluster algebra A of rank n is a commutative ring with unit and no zero divisors, equipped with a distinguished family of generators called cluster variables. The set of cluster variables is the (non-disjoint) union of a distinguished collection of n-subsets called clusters. These clusters have the following exchange property: for any cluster x and any element x ∈ x, there is another cluster obtained from x by replacing x with an element x′ related to x by a binomial exchange relation xx′ = M1 +M2 , (1.1) whereM1 and M2 are two monomials without common divisors in the n−1 variables x − {x}. Furthermore, any two clusters can be obtained from each other by a sequence of exchanges of this kind. The prototypical example of a cluster algebra of rank 1 is the coordinate ring A = C[SL2] of the group SL2, viewed in the following way. Writing a generic element of SL2 as [ a b c d ] , we consider the entries a and d as cluster variables, and the entries b and c as scalars. There are just two clusters {a} and {d}, and A is the algebra over the polynomial ring C[b, c] generated by the cluster variables a and d subject to the binomial exchange relation ad = 1 + bc . Another important incarnation of a cluster algebra of rank 1 is the coordinate ring A = C[SL3/N ] of the base affine space of the special linear group SL3; here N is the maximal unipotent subgroup of SL3 consisting of all unipotent upper triangular matrices. Using the standard notation (x1, x2, x3, x12, x13, x23) for the Plücker coordinates on SL3/N , we view x2 and x13 as cluster variables; then A is the algebra over the polynomial ring C[x1, x3, x12, x13] generated by the two cluster variables x2 and x13 subject to the binomial exchange relation x2x13 = x1x23 + x3x12 .