Kac-Moody groups and cluster algebras

Let Q be a finite quiver without oriented cycles, let \Lambda be the associated preprojective algebra, let g be the associated Kac-Moody Lie algebra with Weyl group W, and let n be the positive part of g. For each Weyl group element w, a subcategory C_w of mod(\Lambda) was introduced by Buan, Iyama, Reiten and Scott. It is known that C_w is a Frobenius category and that its stable category is a Calabi-Yau category of dimension two. We show that C_w yields a cluster algebra structure on the coordinate ring \CC[N(w)] of the unipotent group N(w) := N \cap (w^{-1}N_-w). Here N is the pro-unipotent pro-group with Lie algebra the completion of n. One can identify \CC[N(w)] with a subalgebra of the graded dual of the universal enveloping algebra U(n) of n. Let S^* be the dual of Lusztig's semicanonical basis S of U(n). We show that all cluster monomials of \CC[N(w)] belong to S^*, and that S^* \cap \CC[N(w)] is a basis of \CC[N(w)]. Moreover, we show that the cluster algebra obtained from \CC[N(w)] by formally inverting the generators of the coefficient ring is isomorphic to the algebra \CC[N^w] of regular functions on the unipotent cell N^w := N \cap (B_-wB_-) of the Kac-Moody group G with Lie algebra g. We obtain a corresponding dual semicanonical basis of \CC[N^w]. As one application we obtain a basis for each acyclic cluster algebra, which contains all cluster monomials in a natural way.