N(w)$, we have $$\De_{\varpi_1,w^{-1}(\varpi_1)}(g) = b_0b_2-b_1^2,\qquad
\De_{\varpi_2,w^{-1}(\varpi_2)}(g) = b_1b_3-b_2^2,$$ so $$N^w \cong \left\{\begin{pmatrix}
1 & b_0+b_1z+b_2z^2+b_3z^3 \\
0 & 1 
\end{pmatrix}
\mid b_i\in\C,\ 
(b_0b_2-b_1^2)(b_1b_3-b_2^2) \not = 0
\right\}.$$

Categories of modules over preprojective algebras
=================================================

Seminal works of Ringel and Lusztig have shown that the interaction between Kac-Moody algebras and the representation theory of quivers is essential for understanding the quantum enveloping algebra $U_q(\n)$ and its canonical basis. Following Lusztig, one can also construct the (classical) enveloping algebra $U(\n)$ in terms of the path algebra of a quiver with relations called the preprojective algebra. This will be our basic tool for exploring cluster algebra structures on $\C[N(w)]$ and $\C[N^w]$. In this section we shall introduce the preprojective algebra $\L$ and its nilpotent representations, and explain how it yields an interesting basis of $\C[N]$ dual to Lusztig’s semicanonical basis of $U(\n)$. We will then describe some categories $\CC_w$ of $\L$-modules introduced and studied in general by Buan, Iyama, Reiten and Scott [@BIRS] (and independently in [@GLSUni1], for adaptable Weyl group elements $w$). This will provide a categorical model for the cluster algebras of §\[sect5\].

The preprojective algebra
-------------------------

Let $Q$ be a quiver obtained by orienting the edges of the graph 