Kac-Moody groups and cluster algebras

Understand

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).