Abstract

In this paper, we define a notion of AS-Gorenstein algebra for N-graded algebras, and show that symmetric AS-regular algebras of Gorenstein pa-rameter 1 are exactly preprojective algebras of quasi-Fano algebras. This result can be compared with the fact that symmetric graded Frobenius al-gebras of Gorenstein parameter −1 are exactly trivial extensions of finite dimensional algebras. The results of this paper suggest that there is a strong interaction between classification problems in noncommutative al-gebraic geometry and those in representation theory of finite dimensional algebras.