The Monoid of Families of Quiver Representations

A monoid structure on families of representations of a quiveris introduced by taking extensions of representations in families,that is, subvarieties of the varieties of representations. Thestudy of this monoid leads to interesting interactions betweenrepresentation theory, algebraic geometry and quantum grouptheory. For example, it produces a wealth of interesting examplesof families of quiver representations, which can be analysedby representation-theoretic and geometric methods. Conversely,results from representation theory, in particular A. Schofield'swork on general properties of quiver representations, allowus to relate the monoid to certain degenerate forms of quantizedenveloping algebras. 2000 Mathematical Subject Classification: 16G20, 14L30, 17B37.