Exercise 6, § 1

(a) Let $(M_s)_{s\in Som(Q)}$ be a family of $k$-modules and let $(u_f)_{f\in Fl(Q)}$ be a family where, for every $f\in$ Fl(Q)$,u_f$ is a linear mapping from $M_{o(f)}$ into $M_{t(f)}$. Let M be the direct sum of the family $(M_s)$; for $s\in S$, denote by $j_s$ the canonical injection of $M_s$ into M and by $p_s$ the canonical projection of M onto $M_s$. Prove that there exists a unique structure of $A_Q$-module on M such that one has $(x_s)_M=j_s\circ p_s$ and $(x_f)_M=j_{t(f)}\circ u_f\circ p_{o(f)}$ for every $s\in$ Som(Q) and every $f\in$ Fl(Q).

Conversely, every $A_Q$-module is of this form.

(b) If Λ is a $k$-module and $s\in$ Som(Q), denote by $\Lambda (s)$ the $A_Q$-module associated with the family $(M_i)_{i\in Som(Q)}$ of $k$-modules given by $M_i= \Lambda$ if $i=s$ and $M_i= 0$ otherwise, and with the family $(u_f)_{f\in Fl(Q)}$ of linear mappings, where $u_f$ = 0 for every $f\in$ Fl(Q).

Let Λ and $\Lambda '$ be non-zero $k$-modules. Let $s$ and $s'$ be vertices of Q. Prove that $\Lambda (s)$ is isomorphic to $\Lambda '(s')$ if and only if Λ and $\Lambda '$ are isomorphic $k$-modules and if $s=s'$.

(c) If Λ is a simple $k$-module, then $\Lambda (s)$ is a simple $A_Q$-module for every vertex $s$.

(d) Suppose in addition that every loop in Q is of length zero. Prove that every simple $A_Q$-module is of the form $\Lambda (s)$, where Λ is a simple $k$-module and $s$ is a vertex of Q.