Extension Quiver for Lie Superalgebra $\mathfrak{q}(3)$

We describe all blocks of the category of finite-dimensional $\mathfrak{q}(3)$-supermodules by providing their extension quivers. We also obtain two general results about the representation of $\mathfrak{q}(n)$: we show that the Ext quiver of the standard block of $\mathfrak{q}(n)$ is obtained from the principal block of $\mathfrak{q}(n-1)$ by identifying certain vertices of the quiver and prove a “virtual” BGG-reciprocity for $\mathfrak{q}(n)$. The latter result is used to compute the radical filtrations of $\mathfrak{q}(3)$ projective covers.