Bernstein–Gelfand–Ponomarev reflection functors

Definition

Let $k$ be a field and let $Q$ be a finite quiver. A vertex $i$ is a sink if no arrow has source $i$, and a source if no arrow has target $i$. Let $\sigma_i Q$ be the quiver obtained from $Q$ by reversing every arrow incident to $i$; write $a^*$ for the reverse of $a$. Let $i$ be a sink of $Q$ and $V$ a representation of $Q$ over $k$. The representation $S_i^+ V$ of $\sigma_i Q$ is defined by:

1. $(S_i^+ V)_j = V_j$ and $(S_i^+ V)_b = V_b$ for vertices $j \ne i$ and arrows $b$ not incident to $i$;
2. $(S_i^+ V)_i = \ker \big( \bigoplus_{a \colon j \to i} V_j \xrightarrow{(V_a)} V_i \big)$, the sum taken over the arrows $a$ ending at $i$;
3. for an arrow $a \colon j \to i$, the map $(S_i^+ V)_{a^*} \colon (S_i^+ V)_i \to V_j$ is the inclusion of the kernel followed by the projection onto the summand $V_j$ of index $a$.

With the evident action on morphisms, $S_i^+$ is a functor. Dually, for a source $i$, $S_i^-$ replaces $V_i$ by the cokernel of $V_i \to \bigoplus_{a \colon i \to j} V_j$.