Multiplicative preprojective algebras, middle convolution and the Deligne–Simpson problem

By W. Crawley‐Boevey, P. Shaw. 2004

117 citations recorded by the source corpus.

Abstract 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 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 A 1 A 2 … A k = 1 with the A i in prescribed conjugacy classes in GL n ( C ) .

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bb3b4f9c04f9eee1573272fc5031becd31a8b6b5

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In Algebra and Number Theory

1. I. Reiten
2. Osamu Iyama
3. Aslak Bakke Buan
4. B. Keller
5. J. Scott
6. Carlos Simpson
7. Gustavo Jasso
8. Daisuke Yamakawa

In Geometry and Topology

1. I. Reiten
2. Osamu Iyama
3. Aslak Bakke Buan
4. B. Keller
5. Bernard Leclerc
6. P. Boalch
7. Daisuke Yamakawa
8. J. Scott

In Mathematical Physics

1. I. Reiten
2. P. Boalch
3. Daisuke Yamakawa
4. Osamu Iyama
5. Aslak Bakke Buan
6. Carlos Simpson
7. J. Scott
8. Bernard Leclerc

Citations in the corpus

• Katz's middle convolution algorithm (2006, methodology): “in the situation of Theorem 4.10 for both of the morphisms. We don’t describe the proof here but refer to Katz [86], V¨olklein [139], Dettweiler-Reiter [44], and more recently Crawley-Boevey and Shaw [35]. 4.8. Detecting emptiness of the moduli space. One of the main features of Katz’s construction ”
• Twisted wild character varieties (2015, background): “They also yield a twisted version of the (coloured) multiplicative quiver varieties [23, 43, 19].”
• Cluster structures for 2-Calabi–Yau categories and unipotent groups (2007, methodology): “completions of preprojective algebras of a finite connected quiver Qwith no oriented cycles, and mainly those which are not Dynkin. These algebras Λ are known to be derived 2-CY [B, CB, BBK] (see also [CBS]). When the quiver Qis a (generalized) extended Dynkin quiver, then tiltingΛ-modulesofprojectiv”
• Geometry and braiding of Stokes data; Fission and wild character varieties (2011, methodology): “This is the map used in [27] to reverse the orientation of the edges of multiplicative quiver varieties.”
• Geometry of Multiplicative Preprojective Algebra (2007, methodology): “It was given by Crawley-Boevey and Shaw [11].”