Let $Q=(Q_0,Q_1,s,e)$ be a finite quiver.
In the formal construction of the preprojective algebra associated to $Q$, we set $\ Q^*_1=\{\alpha^*:y\rightarrow x| \alpha\in Q_1, \alpha:x\rightarrow y \}$ and $\overline Q_1=Q^*_1 \cup Q_1$, and we define $\mathscr P(Q)= k \overline Q/<\rho>$, where $k$ is an algebraically closed field and $$\rho=\underset {\alpha\in Q_1} \sum (\alpha\alpha^*-\alpha^*\alpha)$$.
Having this formal construction of the preprojective algebra $\mathscr P(Q)$, associated to the path algebra $kQ$, I was just wondering if for a given bound quiver $(Q,I)$, one could formally construct a similar algebra associated to the bound quiver algebra $kQ/I$, such that the new algebra could be considered as the counterpart of the preprojective algebra associated to $kQ$.
If not, what are the obstacles in the way of this generalization?
I can already think of at least one issue, which is the fact that we may have $kQ/I\simeq kQ/J$, for different admissible ideals $I$ and $J$ in $kQ$. But, is this a formidable issue that we could not tackle, or are there other problems that prohibit the construction?
