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Rigid modules over preprojective algebras II: The Kac-Moody case

Let Q be a finite quiver without oriented cycles, and let L be the associated preprojective algebra. We construct many Frobenius subcategories of mod(L), which yield categorifications of large classes of cluster algebras. This includes all acyclic cluster algebras. We show that all cluster monomials can be realized as elements of the dual of Lusztig's semicanonical basis of a universal enveloping algebra U(n), where n is a maximal nilpotent subalgebra of the symmetric Kac-Moody Lie algebra g associated to the quiver Q.