  We describe recent work on preprojective algebras and moduli spaces of their
representations. We give an analogue of Kac's Theorem, characterizing the
dimension types of indecomposable coherent sheaves over weighted projective
lines in terms of loop algebras of Kac-Moody Lie algebras, and explain how it
is proved using Hall algebras. We discuss applications to the problem of
describing the possible conjugacy classes of sums and products of matrices in
known conjugacy classes.
