Cluster algebra structures for unipotent cells

This a report on joint work with B. Leclerc and Jan Schröer. 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. $C_w$ is a Frobenius category that its stably 2-Calabi-Yau. We show that $C_w$ yields a cluster algebra structure on the coordinate ring $\mathbb{C}[N(w)]$ of the unipotent group $N(w) := N \cap (w^{-1}N_-w)$. One can identify $\mathbb{C}[N(w)]$ 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 $\mathbb{C}[N(w)]$ belong to $S^*$, and that $S^* \cap \mathbb{C}[N(w)]$ is a basis of $\mathbb{C}[N(w)]$. It would be be nice to have a quantum version of these results and ultimately relate quantum cluster monomials with the dual canoncial basis.