We investigate the generalized Kronecker algebra $\mathcal{ K}_r = k\Gamma_r$ with $r \geq 3$ arrows. Given a regular component $\mathcal{ C}$ of the Auslander-Reiten quiver of $\mathcal{ K}_r$, we show that the quasi-rank $rk(\mathcal{ C}) \in \mathbb{ Z}_{ \leq 1}$ can be described almost exactly as the distance $\mathcal{ W}(\mathcal{ C}) \in \mathbb{ N}_0$ between two non-intersecting cones in $\mathcal{ C}$, given by modules with the equal images and the equal kernels property; more precisley, we show that the two numbers are linked by the inequality \[ -\mathcal{ W}(\mathcal{ C}) \leq rk(\mathcal{ C}) \leq - \mathcal{ W}(\mathcal{ C}) + 3.\] Utilizing covering theory, we construct for each $n \in \mathbb{ N}_0$ a bijection $\varphi_n$ between the field $k$ and $\{ \mathcal{ C} \mid \mathcal{ C} \ \text{ regular component}, \ \mathcal{ W}(\mathcal{ C}) = n \}$. As a consequence, we get new results about the number of regular components of a fixed quasi-rank.
Categories: math.RT
