Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 9 (2013), 037, 13 pages      arXiv:1208.3613      http://dx.doi.org/10.3842/SIGMA.2013.037

A Note on the Automorphism Group of the Bielawski-Pidstrygach Quiver

Igor Mencattini and Alberto Tacchella
ICMC - Universidade de São Paulo, Avenida Trabalhador São-carlense, 400, 13566-590 São Carlos - SP, Brasil

Received August 29, 2012, in final form April 26, 2013; Published online April 30, 2013

Abstract
We show that there exists a morphism between a group Γalg introduced by G. Wilson and a quotient of the group of tame symplectic automorphisms of the path algebra of a quiver introduced by Bielawski and Pidstrygach. The latter is known to act transitively on the phase space Cn,2 of the Gibbons-Hermsen integrable system of rank 2, and we prove that the subgroup generated by the image of Γalg together with a particular tame symplectic automorphism has the property that, for every pair of points of the regular and semisimple locus of Cn,2, the subgroup contains an element sending the first point to the second.

Key words: Gibbons-Hermsen system; quiver varieties; noncommutative symplectic geometry; integrable systems.

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References

  1. Baranovsky V., Ginzburg V., Kuznetsov A., Wilson's Grassmannian and a noncommutative quadric, Int. Math. Res. Not. 2003 (2003), 1155-1197, math.AG/0203116.
  2. Berest Y., Wilson G., Automorphisms and ideals of the Weyl algebra, Math. Ann. 318 (2000), 127-147, math.QA/0102190.
  3. Bielawski R., Pidstrygach V., On the symplectic structure of instanton moduli spaces, Adv. Math. 226 (2011), 2796-2824, arXiv:0812.4918.
  4. Bocklandt R., Le Bruyn L., Necklace Lie algebras and noncommutative symplectic geometry, Math. Z. 240 (2002), 141-167, math.AG/0010030.
  5. Dixmier J., Sur les algèbres de Weyl, Bull. Soc. Math. France 96 (1968), 209-242.
  6. Gibbons J., Hermsen T., A generalisation of the Calogero-Moser system, Phys. D 11 (1984), 337-348.
  7. Ginzburg V., Non-commutative symplectic geometry, quiver varieties, and operads, Math. Res. Lett. 8 (2001), 377-400, math.QA/0005165.
  8. Kontsevich M., Formal (non)commutative symplectic geometry, in The Gel'fand Mathematical Seminars, 1990-1992, Birkhäuser Boston, Boston, MA, 1993, 173-187.
  9. Makar-Limanov L.G., On automorphisms of Weyl algebra, Bull. Soc. Math. France 112 (1984), 359-363.
  10. Makar-Limanov L.G., The automorphisms of the free algebra with two generators, Funct. Anal. Appl. 4 (1970), 262-264.
  11. Nagao H., On GL(2,K[x]), J. Inst. Polytech. Osaka City Univ. Ser. A 10 (1959), 117-121.
  12. Nakajima H., Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), 365-416.
  13. Nekrasov N., Schwarz A., Instantons on noncommutative R4, and (2,0) superconformal six-dimensional theory, Comm. Math. Phys. 198 (1998), 689-703, hep-th/9802068.
  14. Serre J.P., Trees, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.
  15. Wilson G., Equivariant maps between Calogero-Moser spaces, arXiv:1009.3660.
  16. Wilson G., Notes on the vector adelic Grassmannian, 2009, unpublished.

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