Cluster algebra structures and semicanonical bases for unipotent groups

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Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. To each terminal representation M of Q (these are certain preinjective representations), we attach a natural subcategory Cₘ of mod(Λ). We show that Cₘ is a Frobenius category,and that its stable category is a Calabi-Yau category of dimension 2. Then we develop a theory of mutations of maximal rigid objects of Cₘ, analogous to the mutations of clusters in Fomin and Zelevinsky's theory of cluster algebras. We show that Cₘ yields a categorification of a cluster algebra A(Cₘ), which is not acyclic in general. We give a realization of A(Cₘ) as a subalgebra of the graded dual of the enveloping algebra U(), where is a maximal nilpotent subalgebra of the symmetric Kac-Moody Lie algebra associated to the quiver Q. Let S* be the dual of Lusztig's semicanonical basis S of U(). We show that all cluster monomials of A(Cₘ) belong to S*, and that S* ∩ A(Cₘ) is a basis of A(Cₘ). Next, we prove that A(Cₘ) is naturally isomorphic to the coordinate ring of the finite-dimensional unipotent subgroup N(w) of the Kac-Moody group G attached to. Here w = w(M) is the adaptable element of the Weyl group of which we associate to each terminal representation M of Q. Moreover, we show that the cluster algebra obtained from A(Cₘ) by formally inverting the generators of the coefficient ring is isomorphic to the coordinate ring of the unipotent cell Nʷ:= N ∩ (B-wB-) of G. We obtain a corresponding dual semicanonical basis of this coorindate ring.