Given a finite dimensional selfinjective algebra $ A$ (given by connected quiver and relations) and a basic non-projective generator $ M$ of mod-A. Let $ B:=End_A(M)$ and $ M_i$ the indecomposable summands of $ M$ . We can also assume that $ B$ is given by quiver and relations. The indecomposable projective B-modules are $ P_i:=Hom_A(M,M_i)$ . Is there a nice way to describe the socle of $ P_i$ ? More specifically I search for a nice criterion when the socle of $ P_i$ is simple, that is dim($ soc(Hom_A(M,M_i))$ )=1. When all socle of the indecomposable projective modules are simple the algebra is called QF-2 algebra, so I wonder whether there exists a useful characterisation when a generator has a QF-2 endomorphism ring. If it helps we can also pose additional assumptions on the selfinjective algebra and the generator.
Here a special case that Im especially interested in: Let $ A$ be a selfinjective Nakayama algebra and $ M:=A \oplus N$ when $ N$ is any indecomposable non-projective $ A$ -module. Then $ B=End_A(M)$ is QF-2 but the only way I can show this is calcuation of the quiver and relations of $ B$ which is not so nice and lengthy.
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