Kac–Moody groups and cluster algebras

Let Q be a finite quiver without oriented cycles, let Λ 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 Cw of mod(Λ) was introduced by Buan, Iyama, Reiten and Scott. It is known that Cw is a Frobenius category and that its stable category is a Calabi–Yau category of dimension two. We show that Cw yields a cluster algebra structure on the coordinate ring CN(w)] of the unipotent group N(w):=N∩(w−1N₋w). Here N is the pro-unipotent pro-group with Lie algebra the completion of n. One can identify CN(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 CN(w)] belong to S^?, and that S^?∩CN(w)] is a C-basis of CN(w)]. Moreover, we show that the cluster algebra obtained from CN(w)] by formally inverting the generators of the coefficient ring is isomorphic to the algebra CNw] of regular functions on the unipotent cell Nw of the Kac–Moody group with Lie algebra g. We obtain a corresponding dual semicanonical basis of CNw]. As one application we obtain a basis for each acyclic cluster algebra, which contains all cluster monomials in a natural way.