Self-injective radical-cube-zero algebras satisfy the finite generation condition (Fg) exactly when their type is Dynkin, extended Dynkin other than ~An, or ~An with a finite-order Nakayama automorphism, assuming the characteristic is good.
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Skew Group Algebras, (Fg) and Self-injective Rad-Cube-Zero Algebras
Self-injective radical-cube-zero algebras satisfy the finite generation condition (Fg) exactly when their type is Dynkin, extended Dynkin other than ~An, or ~An with a finite-order Nakayama automorphism, assuming the characteristic is good.