Kac–Moody groups and cluster algebras

Abstract

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 C w of mod ( Λ ) was introduced by Buan, Iyama, Reiten and Scott. It is known that C w is a Frobenius category and that its stable category C ̲ w is a Calabi–Yau category of dimension two. We show that C w yields a cluster algebra structure on the coordinate ring C [ N ( w ) ] of the unipotent group N ( w ) : = N ∩ ( w − 1 N − w ) . Here N is the pro-unipotent pro-group with Lie algebra the completion n ˆ of n . One can identify C [ N ( w ) ] with a subalgebra of U ( n ) gr ⁎ , the graded dual of the universal enveloping algebra U ( n ) of n . Let S ⁎ be the dual of Lusztigʼs semicanonical basis S of U ( n ) . We show that all cluster monomials of C [ N ( w ) ] belong to S ⁎ , and that S ⁎ ∩ C [ N ( w ) ] is a C -basis of C [ N ( w ) ] . Moreover, we show that the cluster algebra obtained from C [ N ( w ) ] by formally inverting the generators of the coefficient ring is isomorphic to the algebra C [ N w ] of regular functions on the unipotent cell N w of the Kac–Moody group with Lie algebra g . We obtain a corresponding dual semicanonical basis of C [ N w ] . As one application we obtain a basis for each acyclic cluster algebra, which contains all cluster monomials in a natural way.