M&& \_[aE]{} M\_a M\_[a\^\*]{}- M\_[a\^\*]{}M\_a

The group $\G_\alpha(\R)$ acts naturally on both sides, and the map $\mu_\R$ is equivariant. Now denote by $\R\overline{Q}$ the path algebra of $\overline{Q}$ and let $\sigma=\sum_{a\in E} aa^*-a^*a\in \R\overline{Q}$. Then the points of $\mu_\R^{-1}(0)$ corresponds to the locally free representations of the $\R$-algebra $$\Pi_\R(Q)=\R\overline{Q}/(\sigma)$$ of dimension $\alpha$. The $\R$-algebra $\Pi_\R(Q)$ is called the preprojective algebra of $Q$ over $\R$. So $\mu_\R^{-1}(0)/\G_\alpha(\R)$ can be thought as the set of isomorphism classes of locally free representations of $\Pi_\R(Q)$. We define $m_\alpha(\Pi_\R(Q))$ as we defined $m_\alpha(\R Q)$ with $\rep^\alpha(Q,\R)$ replaced by $\mu^{-1}_\R(0)$. We also define $a_\alpha(\Pi_\R(Q))$ similarly as $a_\alpha(\R Q)$ with $\rep^\alpha_{\rm a.i.}(Q,\R)$ replaced by the set of absolutely indecomposable locally free representations of $\Pi_\R(Q)$. Then we have

\[main2\] If $\R$ is a Frobenius algebra over a finite or algebraically closed field $\k$ then we have $$m_\alpha(\Pi_\R(Q))=m_\alpha(\R[\epsilon] Q).$$If moreover $\R$ is local, split and if $\k$ contains a primitive $|\alpha|$-th root of unity then $$a_\alpha(\Pi_\R(Q))=a_\alpha(\R[\epsilon] Q).$$ \[maintheoquiv\]

When $\R=\k$ is a field this result relates representations of a quiver over $
\k_2=\k[\epsilon]$ and its preprojective algebra. Thus it unexpectedly bridges two papers [@GLS1; @GLS] of Geiss–Leclerc–Schröer. Also in the case of the Jordan quiver and $\R=\k$ a finite 