On the Returning Arrows of Graded Self-injective Algebras

摘要: A translation quiver $Q$ is a with mono map $\tau$, called translation, defined on subset of the vertices. Recently, varies construction gives rise to obtained from original one by inserting returning arrows, that is, arrows $x$ $ \tau x for certain vertices $x$. In this paper, we show such phenomenon also appear in Gabriel quivers graded self-injective algebras when construct trivial extension, using automorphism induced Nakayama functor as translation. We under condition, two numerical invariants, complexity and representation dimension, increase appear.