A quiver representation over a field 
assigns a vector space 
to each vertex of a quiver
and a linear map 
to each arrow 
. A morphism is a family of linear maps 
commuting with every arrow
map. For finite-dimensional representations of a finite quiver, its dimension vector
is 
and the representation space for fixed vertex spaces is
 |
The group 
acts by changing bases at the vertices. A one-loop quiver
represents one endomorphism, and its isomorphisms
are similarity transformations. Framing adds fixed vector spaces and maps involving
them in the construction of Nakajima quiver
varieties.
