Returning Arrows for Self-injective Algebras and Artin-Schelter Regular Algebras

In this paper, we discuss returning arrows with respect to the Nakayama translation appearing in the quivers of some important algebras when we construct extensions. When constructing twisted trivial extensions for a graded self-injective algebra, we show that the returning arrows appear in the quiver, that the complexity increases by 1 in Koszul cases, and the representation dimension also increases by 1 under certain additional conditions. By applying Koszul duality, for each Koszul Artin-Schelter regular algebra of global dimension l and Gelfand-Kirilov dimension $c$, we construct a family of Koszul Artin-Schelter regular algebras of global dimension $l+1$ and Gelfand-Kirilov dimension $c+1$, among them one is central extension and one is $l+1$-Calabi-Yau.