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

EXAMPLES.
(1) If Q consists of one vertex and one loop, then kQ k[T]. If Q has one
vertex and r loops, then kQ is the free associative algebra on r letters.
(2) If there is at most one path between any two points, then kQ can be
identified with the subalgebra
{C ¡ M (k) ¢ C =0 if no path from j to i}
n ij
of M (k). If Q is 1 ¡ 2 ¡ ... ¡ n this is the lower triangular matrices.
n
(i£
IDEMPOTENTS. Set A=kQ.
(1) The e
i
are orthogonal idempotents, ie e e
i j
2
= 0 j), e
i
= e .
i
n
(2) A has an identity given by 1 = ¤ e .
i=1 i
(3) The spaces Ae , e A, and e Ae have as bases the paths starting at i
i j j i
and/or terminating at j.
n
(4) A = ¥ Ae , so each Ae is a projective left A-module.
i=1 i i
(5) If X is a left A-module, then Hom (Ae ,X) e X.
A i i
(6) If 0£ f¡ Ae and 0£ g¡ e A then fg£ 0.
i i
PROOF. Look at the longest paths x,y involved in f,g. In the product fg the
coefficient of xy cannot be zero.
(7) The e are primitive idempotents, ie Ae is a indecomposable module.
i i
2
PROOF. If End (Ae ) e Ae contains idempotent f, then f =f=fe , so
A i i i i
f(e -f)=0. Now use (6).
i
(8) If e ¡ Ae A then i=j.
i j
PROOF. Ae A has as basis the paths passing through the vertex j.
j
(9) The e are inequivalent, ie Ae ¦ Ae for i£ j.
i i j
PROOF. Thanks to (5), inverse isomorphisms give elements f¡ e Ae , g¡ e Ae
i j j i
with fg=e and gf=e . This contradicts (8).
i j
4