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

¨
u © ¢ 0
0 v¢
which corresponds to U¥ V.
Finally Hom( ¢
so
,U) gives an exact sequence
f 1
0 ¡ Hom(V,U) ¡ Hom(X,U) ¡ Hom(U,U) ¡ Ext (V,U),
dim Hom(V,U) - dim Hom(X,U) + dim Hom(U,U) - dim Im(f) = 0,
but f(1 )= ¢
U
CONSEQUENCES.
£ 0, so dim Hom(X,U) £ dim Hom(U¥ V,U), and hence X ¦ U¥ V.
(1) If O is an orbit Rep(¤ in ) of maximal dimension, X=U¥ and V, then
X
1
Ext (V,U)=0.
A
PROOF. If there is non-split extension ¡ 0 ¡ U ¡ E ¡ V 0 then O O \O , so
X E E
dim O < dim O .
X E
(2) If O is closed then X is semisimple.
X
REMARKS.
(1) Suppose Q has no oriented cycles. Let z¡ Rep(¤ ) be the element with all
matrices z =0. We can easily show that z is in the closure of every orbit,
¢
and it follows that there are no non-constant polynomial invariants.
Moreover, an orbit O is closed X is semisimple, for the only semisimple
X
module of dimension ¤ is R(z), and {z} is clearly a closed orbit.
(2) If Q is allowed to have oriented cycles, x,x¥ ¡ Rep(¤ ) and R(x) and R(x¥ )
are non-isomorphic semisimple modules, then there is a polynomial invariant
§ (of the form f ) with § (x)£ § (y). In case Q has only one vertex and
ai
char k=0 this is proved in §12.6 of M.Artin, On Azumaya algebras and finite
dimensional representations of rings, J.Algebra 11 (1969), 532-563, but it
seems to be true in general. It follows that O is closed X is
X
semisimple.
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