  A monoid structure on families of representations of a quiver is introduced
by taking extensions of representations in families, i.e. subvarieties of the
varieties of representations. The study of this monoid leads to interesting
interactions between representation theory, algebraic geometry and quantum
group theory. For example, it produces a wealth of interesting examples of
families of quiver representations, which can be analyzed by representation
theoretic and geometric methods. Conversely, results from representation
theory, in particular A. Schofield's work on general properties of quiver
representations, allow us to relate the monoid to certain degenerate forms of
quantized enveloping algebras.
