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I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory by Ibrahim Assem, Daniel Simson, Andrzej Skowronski.

Let $A$ be a $K$-algebra and $T_A$ be a tilting module. Let $B=\operatorname{End}_A(T_A)$. Let $D$ be the standard dual. That is $D(M)=\operatorname{Hom}_{K}(M, K)$.

I have question on page 211, Example 3.11. On line 4 of Example 3.11 on page 211, it is said that the ordinary quiver of $B = \operatorname{End}_A(T_{A})$ is $\circ \overset{\mu}{\leftarrow} \circ \overset{\lambda}{\leftarrow} \circ $, $\lambda \mu =0$. How to compute the ordinary quiver of $B = \operatorname{End}_A(T_{A})$ from the first diagram in Example 3.11 directly? Thank you very much.

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  • $\begingroup$ could you edit this so it has the correct title of the book and the authors' names? Thanks. $\endgroup$ – Matthew Towers Jul 6 '13 at 12:45
  • $\begingroup$ @mt_, thank you very much. I have edited. $\endgroup$ – LJR Jul 6 '13 at 13:42
  • $\begingroup$ I have written a detailed work through for precisely this example way back, in a question with probably obscure title: math.stackexchange.com/questions/142180/… (see the second answer) $\endgroup$ – Aaron Jul 8 '13 at 21:56
  • $\begingroup$ @Aaron, thank you very much. $\endgroup$ – LJR Jul 9 '13 at 1:35
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The module $T$ has three summands, namely the simple modules at the initial and terminal vertices and the projective with length three. So End(T) has three idempotents, one for each summand, which correspond to the vertices of the quiver given.

There are two obvious morphisms between summands of $T$: inclusion of one simple as the socle of the projective, and a surjection of the projective onto the other simple. These give you the maps $\mu$ and $\lambda$, and together with the identity maps on summands they give all possible morphisms between summands of $T$. To see this, label the simples 1,2,3 and note that the Loewy structure of the projective is

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