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Let $Q$ be a ADE type quiver and $s_i$ ($i$ runs through the vertices of $Q$) be the simple $\Lambda$-module with 1-dimensional vector space at vertex $i$ and zero-dim at other vertices. Here $\Lambda$ is the preprojective algebra of $Q$.

In Geiss, Leclerc, and Schröer's paper, they gave a geometric realization of irreducible $\lambda$ highest weight module by considering constructible function on the variety of nilpotent $\Lambda$-module which are isormorphic to submodule of $q_\lambda$. Here $q_\lambda$ is a direct sum of some $q_i$, where $q_i$ is the injective hull of $s_i$.

I want to calculate some small examples which are mainly about ADE type quiver. Are there any reference finding the injective hull $q_i$ of simple module $s_i$? Thank you so much.

EDIT: I have found a reference. In Alistair Savage, Peter Tingley's paper:Quiver grassmannians, quiver varieties and the preprojective algebra definition 2.7. Assume $Q$ is a quiver of finite type. For any vertex $i$, let $$q^i=\text{Hom}_{\mathbb{C}}(e_i\mathcal{P},\mathbb{C}).$$The left $\mathcal{P}$-module structure on $q^i$ is by setting $a\cdot f(x)=f(xa)$.

EDIT 2: I have found a way to draw $q^i$ in type A. The next thing I want to know is about indecomposable module. For ADE type quiver, what is the injective hull of an indecomposable module $M$ over the preprojective algebra $\mathcal{P}$? Is its injective hull equal to its socle?

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The preprojective algebra of an ADE graph is non-symmetric Frobenius, so the injective hull of $s_i$ is isomorphic to the indecomposable projective module $$\mathcal P e_{\phi(i)},$$ where $\phi$ is the inverse of the Nakayama permutation.

For preprojective algebras of ADE graphs the Nakayama permutation is either the identity or it has order two, and it is given as follows.

In type $\mathsf {A_n}$: the unique graph automorphism of order two.

In type $\mathsf {D_n}$: the identity if $n$ is even and the unique graph automorphism of order two if $n$ is odd.

In type $\mathsf {E_6}$: the unique graph automorphism of order two.

In type $\mathsf {E_7}$ and $\mathsf {E_8}$: the identity.

Sources:

Karin Erdmann and Nicole Snashall, On Hochschild cohomology of preprojective algebras. I, II, J. Algebra 205 (1998), no. 2, 391--412, 413--434.

Erdmann, Karin; Snashall, Nicole. Preprojective algebras of Dynkin type, periodicity and the second Hochschild cohomology. Algebras and modules, II (Geiranger, 1996), 183--193, CMS Conf. Proc., 24, Amer. Math. Soc., Providence, RI, 1998

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  • $\begingroup$ Thank you. This is easier to work without the dull. Do you know how to find the injective hull of an indecomposable module? Is it the injecive hull of the scole of that module? I am not so sure. $\endgroup$ – Ben Mar 24 '16 at 6:08
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    $\begingroup$ Yes, that is correct. $\endgroup$ – Dag Oskar Madsen Mar 24 '16 at 10:01
  • $\begingroup$ I think this my last question: do you have any reference for this fact? Is this something standard and not only true for ADE type preprojective algebra? Thank you! $\endgroup$ – Ben Mar 24 '16 at 12:51
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    $\begingroup$ It is a standard result for finite dimensional algebras. The essential property is that the socle of a finitely generated module is zero if and only the module is zero. See for instance Auslander, Reiten, Smalø: Representation theory of Artin algebras, Proposition II.4.1. $\endgroup$ – Dag Oskar Madsen Mar 24 '16 at 13:01

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