Lectures on Representations of Quivers by William Crawley-Boevey ...

INDECOMPOSABLE PROJECTIVES AND INJECTIVES.
¥
(1) The modules P(i) = Ae
i
are a complete set of non-isomorphic
indecomposable projective left A-modules.
PROOF. The e
i
are inequivalent primitive idempotents and A =
n
Ae . Now
i=1 i
use Krull-Schmidt.
(2) The modules I(i) = (P(i)) = D(e A) are a complete set of
i
non-isomorphic indecomposable injective left A-modules.
PROOF. Use Hom(-,A) and D.
(3) = ¤ = for any ¤ .
i
PROOF. If X has dimension ¤ , then
1
= dim Hom(P(i),X) - dim Ext (P(i),X) = dim e X = ¤ .
i i
= dim Hom(X,I(i)) = dim Hom(P(i),X) = ¤ .
i
n n
(4) The vectors dim P(i) are a basis of ¦ . The dim I(i) are a basis of ¦ .
PROOF. The module S(i) with dimension vector has a projective resolution
i
0 ¡ P ¡ P ¡ S(i) ¡ 0 and an injective resolution 0 ¡ S(i) ¡ I ¡ I ¡ 0.
1 0 0 1
c:¦ ¡ ¦
COXETER TRANSFORMATION.
n
(1) There is an automorphism
n
with dim P = - c(dim P) for P
projective.
PROOF. Define c via c(dim P(i)) = - dim I(i).
(2) If X is indecomposable and non-projective then dim X = c(dim X).
PROOF. Let 0 ¡ P ¡ Q ¡ X ¡ 0 be a projective resolution. We have an exact
sequence 0 ¡ X ¡ P ¡ Q ¡ 0 and so
dim X = dim P - dim Q = -c(dim P - dim Q) = -c(dim X).
(3) = - = .
PROOF. = = -.
(4) c¤ =¤ ¢¤ ¡ rad(q).
PROOF. =-=(¥ ,¤ ).
REMARK. When is written as a product of reflections, one sees that the
Coxeter transformation is a Coxeter element in the sense of Coxeter groups.
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