$\C[N]$ can be regarded as a kind of (dual) Hall algebra of the category $\nil(\L)$. Indeed, the multiplication of $\C[N]$ encodes extensions in $\nil(\L)$, as shown by the following crucial result. Before stating it, we recall that $\nil(\L)$ possesses a remarkable symmetry with respect to extensions, namely, $\Ext^1_\L(X,Y)$ is isomorphic to the dual of $\Ext^1_\L(Y,X)$ functorially in $X$ and $Y$ (see [@GLS4]). In particular $\dim\Ext^1_\L(X,Y) = \dim \Ext^1_\L(Y,X)$ for every $X, Y$.

\[Th\_mult\] Let $X, Y \in \nil(\L)$.

-   We have $\vph_X \vph_Y = \vph_{X\oplus Y}$.

-   Assume that $\dim \Ext^1_\L(X,Y)=1$, and let $$0\to X \to L \to Y \to 0 
     \quad\mbox{and}\quad 
     0\to Y \to M \to X \to 0$$ be non-split short exact sequences. Then $\vph_X \vph_Y = \vph_L + \vph_M$.

We illustrate Theorem \[Th\_mult\] (ii) in type $A_2$. Take $X=S_1$ and $Y=S_2$. Then we have the non-split short exact sequences $$0\to S_1 \to P_2 \to S_2 \to 0 
 \quad\mbox{and}\quad 
 0\to S_2 \to P_1 \to S_1 \to 0,$$ which imply the relation $$\vph_{S_1} \vph_{S_2} = \vph_{P_2} + \vph_{P_1},$$ that is, the elementary determinantal relation $$D_{1,2}D_{2,3} = D_{1,3} + D_{12,23}$$ on the unitriangular subgroup of $\SL(3,\C)$.

The dual semicanonical basis
----------------------------

We can now introduce the dual semicanonical basis of 