Lectures on Representations of Quivers by William Crawley-Boevey ...
Lectures on Representations of Quivers by William Crawley-Boevey ...
Lectures on Representations of Quivers by William Crawley-Boevey ...
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¨<br />
u © ¢ 0<br />
0 v¢<br />
which corresp<strong>on</strong>ds to U¥ V.<br />
Finally Hom( ¢<br />
so<br />
,U) gives an exact sequence<br />
f 1<br />
0 ¡ Hom(V,U) ¡ Hom(X,U) ¡ Hom(U,U) ¡ Ext (V,U),<br />
dim Hom(V,U) - dim Hom(X,U) + dim Hom(U,U) - dim Im(f) = 0,<br />
but f(1 )= ¢<br />
U<br />
CONSEQUENCES.<br />
£ 0, so dim Hom(X,U) £ dim Hom(U¥ V,U), and hence X ¦ U¥ V.<br />
(1) If O is an orbit Rep(¤ in ) <strong>of</strong> maximal dimensi<strong>on</strong>, X=U¥ and V, then<br />
X<br />
1<br />
Ext (V,U)=0.<br />
A<br />
PROOF. If there is n<strong>on</strong>-split extensi<strong>on</strong> ¡ 0 ¡ U ¡ E ¡ V 0 then O O \O , so<br />
X E E<br />
dim O < dim O .<br />
X E<br />
(2) If O is closed then X is semisimple.<br />
X<br />
REMARKS.<br />
(1) Suppose Q has no oriented cycles. Let z¡ Rep(¤ ) be the element with all<br />
matrices z =0. We can easily show that z is in the closure <strong>of</strong> every orbit,<br />
¢<br />
and it follows that there are no n<strong>on</strong>-c<strong>on</strong>stant polynomial invariants.<br />
Moreover, an orbit O is closed X is semisimple, for the <strong>on</strong>ly semisimple<br />
X<br />
module <strong>of</strong> dimensi<strong>on</strong> ¤ is R(z), and {z} is clearly a closed orbit.<br />
(2) If Q is allowed to have oriented cycles, x,x¥ ¡ Rep(¤ ) and R(x) and R(x¥ )<br />
are n<strong>on</strong>-isomorphic semisimple modules, then there is a polynomial invariant<br />
§ (<strong>of</strong> the form f ) with § (x)£ § (y). In case Q has <strong>on</strong>ly <strong>on</strong>e vertex and<br />
ai<br />
char k=0 this is proved in §12.6 <strong>of</strong> M.Artin, On Azumaya algebras and finite<br />
dimensi<strong>on</strong>al representati<strong>on</strong>s <strong>of</strong> rings, J.Algebra 11 (1969), 532-563, but it<br />
seems to be true in general. It follows that O is closed X is<br />
X<br />
semisimple.<br />
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