LAUSR

AbstractWe investigate the generalized Kronecker algebra ????r = kΓr with r ≥ 3 arrows. Given a regular component ???? of the Auslander-Reiten quiver of ????r, we show that the quasi-rank rk(????) ∈ ℤ≤1 can be described almost exactly as the distance ????(????) ∈ ℕ0 between two non-intersecting cones in ????, 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 −W(C)≤rk(C)≤−W(C)+3.$$-\mathcal{W}(\mathcal{C}) \leq \text{rk}(\mathcal{C}) \leq - \mathcal{W}(\mathcal{C}) + 3.$$Utilizing covering theory, we construct for each n ∈ ℕ0 a bijection φn between the field k and {????∣???? regular component, ????(????) = n}. As a consequence, we get new results about the number of regular components of a fixed quasi-rank.