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associated graded algebra; $QF$ algebra; $QF$-3 algebra; upper Loewy series; lower Loewy series
We prove that an associated graded algebra $R_G$ of a finite dimensional algebra $R$ is $QF$ (= selfinjective) if and only if $R$ is $QF$ and Loewy coincident. Here $R$ is said to be Loewy coincident if, for every primitive idempotent $e$, the upper Loewy series and the lower Loewy series of $Re$ and $eR$ coincide. \endgraf $QF$-3 algebras are an important generalization of $QF$ algebras; note that Auslander algebras form a special class of these algebras. We prove that for a Loewy coincident algebra $R$, the associated graded algebra $R_G$ is $QF$-3 if and only if $R$ is $QF$-3.
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