Generic Variables in Acyclic Cluster Algebras

Abstract

Let $Q$ be an acyclic quiver. We introduce the notion of generic variables for the coefficient-free acyclic cluster algebra $\mathcal{A}(Q)$. We prove that the set $\mathcal{G}(Q)$ of generic variables contains naturally the set $\mathcal{M}(Q)$ of cluster monomials in $\mathcal{A}(Q)$ and that these two sets coincide if and only if $Q$ is a Dynkin quiver. We establish multiplicative properties of these generic variables analogous to multiplicative properties of Lusztig's dual semicanonical basis. This allows to compute explicitly the generic variables when $Q$ is a quiver of affine type. When $Q$ is the Kronecker quiver, the set $\mathcal{G}(Q)$ is a $\mathbb{Z}$-basis of $\mathcal{A}(Q)$ and this basis is compared to Sherman-Zelevinsky and Caldero-Zelevinsky bases.

arXiv comment20 pages. This is an adaptation of the first part of the preprint arXiv:0811.2909. To appear in the Journal of Pure and Applied Algebra