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

2006/04/12 by William Crawley-Boevey, Crawley-Boevey, William · 2 citations
Mathematics · #14H60 #15A24 #16G20 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.math/0604273

openalex publication_date 2006/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Geometry of the Moment Map for Representations of Quivers
• Absolutely indecomposable representations and Kac-Moody Lie algebras (with an appendix by Hiraku Nakajima)
• Unzerlegbare Darstellungen I
• Necklace Lie algebras and noncommutative symplectic geometry
• Decomposition of Marsden–Weinstein Reductions for Representations of Quivers

Cited by

• The Deligne-Simpson Problem
• Introduction to coherent sheaves on weighted projective lines

Related

• Kac's Theorem for weighted projective lines
• Weighted projective lines as fine moduli spaces of quiver representations
• Introduction to quiver varieties -- for ring and representation theoriests
• On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero
• Absolutely indecomposable representations and Kac-Moody Lie algebras (with an appendix by Hiraku Nakajima)