Quiver algebras, weighted projective lines, and the Deligne-Simpson problem

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.

To appear in Proceedings of ICM 2006 Madrid (11 pages)

References in corpus (3)

• Absolutely indecomposable representations and Kac-Moody Lie algebras (with an appendix by Hiraku Nakajima)
• Quiver varieties and a non-commutative P^2
• Geometric construction of representations of affine algebras

Cited by in corpus (7)

• Mirrors of 3d Sicilian theories
• Introduction to coherent sheaves on weighted projective lines
• Quantized multiplicative quiver varieties
• Quivers and difference Painleve equations
• Poisson maps between character varieties: gluing and capping
• Families of Hitchin systems and N=2 theories
• Connections for weighted projective lines