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

§3. The variety of representations
In this section Q is a quiver and A=kQ. We define the variety of
n
representations of Q of dimension vector ¤ ¡ £ , and describe some elementary
properties. We use elementary dimension arguments from algebraic geometry.
The properties we need are listed below.
¡
ALGEBRAIC GEOMETRY.
r
is affine r-space with the Zariski topology. We consider locally closed
subsets U in
r ¡ , ie subsets U which are open in their closure U.
A non-empty locally closed subset U is irreducible if any non-empty subset
of U which is open in U, is dense in U. The space
r ¡
is irreducible.
The dimension of a non-empty locally closed subset U is
sup{n ¢ ¢ Z Z ... Z irreducible subsets closed in U}.
0 1 n
We have dim U = dim U; if W=U V then dim W = max{dim U,dim V}; the space ¡
has dimension r.
r
If an algebraic group G acts on ¡ , then the orbits O are locally closed;
O\O is a union of orbits of dimension strictly smaller than dim O; and if
x¡ O then dim O=dim G-dim Stab (x).
G
n
DEFINITIONS. Let Q be a quiver and ¤ ¡ £ . We define
) t(¢ )
Rep(¤ ) = ¡ Hom (k ,k ).
s(¢
¡ Q k
1 ¢
r
This is isomorphic to ¡
where r = ¤ ¤ ¤ .
¡ t(¢ ¢ s(¢
Rep(¤ ¤ x¡
¢ i¤ ¡ ¢ ¢ 1¤
Q1 ) )
An element ) gives a representation R(x) of Q with R(x)
i
i
= k for
n, and R(x) = x for Q .
1
GL(¤ ¡ GL(¤ We define ) =
n
,k). This is open in
s n 2
¡
¤ where s = ¤ i=1 i i=1
.
i
11
r