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Motivic Donaldson–Thomas invariants and the Kac conjecture

Part of: Projective and enumerative geometry Representation theory of rings and algebras

Published online by Cambridge University Press:  28 February 2013

Sergey Mozgovoy
Sergey Mozgovoy*
Affiliation:
Mathematical Institute, University of Oxford, 24–29 St Giles’, Oxford, OX1 3LB, UK email mozgovoy@maths.ox.ac.uk
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Abstract

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We derive some combinatorial consequences from the positivity of Donaldson–Thomas invariants for symmetric quivers conjectured by Kontsevich and Soibelman and proved recently by Efimov. These results are used to prove the Kac conjecture for quivers having at least one loop at every vertex.

Keywords

Donaldson–Thomas invariantsKac conjecture

MSC classification

Secondary: 16G20: Representations of quivers and partially ordered sets 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 3 , March 2013 , pp. 495 - 504
Copyright
© The Author(s) 2013 

References

Crawley-Boevey, W. and Van den Bergh, M., Absolutely indecomposable representations and Kac–Moody Lie algebras (with an appendix by H. Nakajima), Invent. Math. 155 (2004), 537–559.Google Scholar
Efimov, A. I., Cohomological Hall algebra of a symmetric quiver, Compositio Math. 148 (2012), 1133–1146.CrossRefGoogle Scholar
Getzler, E., Mixed Hodge structures of configuration spaces, Preprint 96-61 (1996), Max Planck Institute for Mathematics, Bonn, arXiv:alg-geom/9510018.Google Scholar
Hua, J., Counting representations of quivers over finite fields, J. Algebra 226 (2000), 1011–1033.CrossRefGoogle Scholar
Kac, V. G., Root systems, representations of quivers and invariant theory, in Invariant theory Montecatini, 1982, Lecture Notes in Mathematics, vol. 996 (Springer, Berlin, 1983), 74–108.Google Scholar
Kontsevich, M. and Soibelman, Y., Stability structures, motivic Donaldson–Thomas invariants and cluster transformations, Preprint (2008), arXiv:0811.2435.Google Scholar
Kontsevich, M. and Soibelman, Y., Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants, Commun. Number Theory Phys. 5 (2011), 231–352.CrossRefGoogle Scholar
Mozgovoy, S., Fermionic forms and quiver varieties, Preprint (2006), arXiv:math/0610084.Google Scholar
Mozgovoy, S., A computational criterion for the Kac conjecture, J. Algebra 318 (2007), 669–679.CrossRefGoogle Scholar
Mozgovoy, S. and Reineke, M., On the number of stable quiver representations over finite fields, J. Pure Appl. Algebra 213 (2009), 430–439.CrossRefGoogle Scholar
Reineke, M., The Harder–Narasimhan system in quantum groups and cohomology of quiver moduli, Invent. Math. 152 (2003), 349–368.CrossRefGoogle Scholar
Reineke, M., Counting rational points of quiver moduli, Int. Math. Res. Not. 2006 (2006), doi:10.1155/IMRN/2006/70456.Google Scholar
Reineke, M., Degenerate cohomological Hall algebra and quantized Donaldson–Thomas invariants for m-loop quivers, Doc. Math. 17 (2012), 1–22.CrossRefGoogle Scholar
Rodriguez-Villegas, F., A refinement of the A-polynomial of quivers, Preprint (2011), arXiv:1102.5308.Google Scholar