2004/04/08 by William Crawley-Boevey, Peter Shaw, Crawley-Boevey, William +1 · 3 citations
Computer Science · Mathematics · #15A24 #16G20 (Primary) 34M50 (Secondary) #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Matrix Theory and Algorithms #Rings and Algebras (math.RA) #math.RA #msc:15A24 #msc:16G20 #msc:34M50
paper · pdf · doi:10.48550/arxiv.math/0404186
21 pages; Lemma 8.1 improved
openalex publication_date 2004/04/08 · arxiv created 2004/07/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a family of algebras which are multiplicative analogues of preprojective algebras, and their deformations, as introduced by M. P. Holland and the first author. We show that these algebras provide a natural setting for the 'middle convolution' operation introduced by N. M. Katz in his book 'Rigid local systems', and put in an algebraic setting by M. Dettweiler and S. Reiter, and by H. Volklein. We prove a homological formula relating the dimensions of Hom and Ext spaces, study varieties of representations of multiplicative preprojective algebras, and use these results to study simple representations. We apply this work to the Deligne-Simpson problem, obtaining a sufficient (and conjecturally necessary) condition for the existence of an irreducible solution to the equation A1 A2 ... Ak = 1 with the Ai in prescribed conjugacy classes in GLn(C).