# Quiver algebras, weighted projective lines, and the Deligne–Simpson problem

### William Crawley-Boevey

University of Leeds, United Kingdom

A subscription is required to access this book chapter.

## Abstract

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.