Returning arrows for self-injective algebras and Artin–Schelter regular algebras

We show that when constructing twisted trivial extensions for a graded self-injective algebra, the returning arrows appear in the quiver, and in the meantime the complexity increases by one in Koszul cases, and the representation dimension also increases by one under certain additional conditions. We apply our results to Koszul Artin–Schelter regular algebras and obtain a family of extensions of a Koszul Artin–Schelter regular algebra, among them one is central extension and one is Calabi–Yau.