Moreover, it is a nice exercise to verify that the number of indecomposable $\L$-modules is then equal to the number of cluster variables. This suggests a close relationship in general between $\L$ and $\C[N]$. To describe it we start with Lusztig’s Lagrangian construction of the enveloping algebra $U(\n)$ [@Lu0; @Lu1]. This is a realization of $U(\n)$ as an algebra of $\C$-valued constructible functions over the varieties of nilpotent representations of $\L$.

Denote by $S_i \ (1\le i\le n)$ the one-dimensional $\L$-module supported on the vertex $i$ of $\overline{Q}$. Given a sequence $\i=(i_1,\ldots,i_d)$ and a nilpotent $\L$-module $X$ of dimension $d$, we introduce the variety $\F_{X,\i}$ of flags of submodules $$\f = (0=F_0\subset F_1 \subset \cdots \subset F_d = X)$$ such that $F_k/F_{k-1} \cong S_{i_k}$ for $k=1,\ldots,d$. This is a projective variety. Denote by $\L_\d$ the affine variety of nilpotent $\L$-modules $X$ with a given dimension vector $\d=(d_i)$, where $\sum_i d_i =d$. Consider the constructible function $\chi_\i$ on $\L_\d$ given by $$\chi_\i(X) = \chi(\F_{X,\i})$$ where $\chi$ denotes the Euler-Poincaré characteristic. Let $\M_{\d}$ be the $\C$-vector space spanned by the functions $\chi_\i$ for all possible sequences $\i$ of length $d$, and let $$\M = \bigoplus_{\d\in\N^n} \M_\d.$$ Lusztig has endowed $\M$ with an 