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General representations of quivers Lecture Notes for Summer School

Summary: General representations of quivers
Lecture Notes for Summer School
Geometry of quiver­representations and preprojective algebras
(10--17 September 2000, Isle of Thorns, Sussex, UK)
Anne­Marie Aubert, Eddy Godelle and Alexander Zimmerman
Ecole Normale Sup'erieure and Universit'e de Picardie
October 30, 2000
A major goal in the representation theory of quivers is to classify all the representations of a given
quiver. The first step of this process was taken by V. Kac who showed that the dimension vectors of
indecomposable representations are precisely the positive roots of a the Kac­Moody Lie algebra associated
to the underlying graph of the quiver (see [K1]). In the course of this investigation, he noted that the
representations of a dimension vector ff of a given quiver Q are parametrized by a vector space Rep(ff) on
which the group GL(ff) acts such that the orbits of the group on the vector space are in 1--1 correspondence
with the isomorphism classes of representations. Certain properties of the representations associated to
a given point of Rep(ff) will be independent of the point chosen in an open subset of Rep(ff) and we
will then say that these properties are true for the general representation. In particular, if R x is the
representation associated to the point x and R x '
i R i;x where R i;x are indecomposable representations,


Source: Aubert, Anne-Marie - Institut de Mathématiques de Jussieu


Collections: Mathematics