Advertisement

Letters in Mathematical Physics

, Volume 107, Issue 7, pp 1265–1291 | Cite as

Poisson–Nijenhuis structures on quiver path algebras

  • Claudio Bartocci
  • Alberto Tacchella
Article
  • 94 Downloads

Abstract

We introduce a notion of noncommutative Poisson–Nijenhuis structure on the path algebra of a quiver. In particular, we focus on the case when the Poisson bracket arises from a noncommutative symplectic form. The formalism is then applied to the study of the Calogero–Moser and Gibbons–Hermsen integrable systems. In the former case, we give a new interpretation of the bihamiltonian reduction performed in Bartocci et al. (Int Math Res Not 2010:279–296, 2010. arXiv:0902.0953).

Keywords

Quiver representations Bihamiltonian structures Integrable systems Calogero-Moser system 

Mathematics Subject Classification

16G20 53D17 70H06 

Notes

Acknowledgements

This work was partially supported by the PRIN “Geometria delle varietà algebriche” and by the University of Genoa’s research Grant “Aspetti matematici della teoria dei campi interagenti e quantizzazione per deformazione.” The authors are grateful to the referee, whose suggestions have been incorporated in Sect. 4. A.T. would like to thank the Department of Mathematics at the University of Genoa for the kind hospitality during the period in which this paper was written.

References

  1. 1.
    Aniceto, I., Avan, J., Jevicki, A.: Poisson structures of Calogero–Moser and Ruijsenaars-Schneider models. J. Phys. A Math. Theor. 43, 185–201 (2010). arXiv:0912.3468 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Avan, J., Ragoucy, E.: Rational Calogero–Moser model: explicit form and r-matrix of the second Poisson structure. SIGMA 8, 079 (2012). arXiv:1207.5368 MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bartocci, C., Falqui, G., Mencattini, I., Ortenzi, G., Pedroni, M.: On the geometric origin of the bi-Hamiltonian structure of the Calogero–Moser system. Int. Math. Res. Not. 2010, 279–296 (2010). arXiv:0902.0953 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bielawski, R.: Quivers and Poisson structures. Manuscr. Math. 141, 29–49 (2013). arXiv:1108.3222 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bielawski, R., Pidstrygach, V.: On the symplectic structure of instanton moduli spaces. Adv. Math. 226, 2796–2824 (2011). arXiv:0812.4918 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bocklandt, R., Le Bruyn, L.: Necklace Lie algebras and noncommutative symplectic geometry. Math. Z. 240, 141–167 (2002). arXiv:math/0010030
  7. 7.
    Brion, M.: Representations of quivers. In: Geometric Methods in Representation Theory. I, vol. 24 of Sémin. Congr. Soc. Math. France, Paris, pp. 103–144 (2012)Google Scholar
  8. 8.
    Cuntz, J., Quillen, D.: Algebra extensions and nonsingularity. J. Am. Math. Soc. 8, 251–289 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Donagi, R., Witten, E.: Supersymmetric Yang-Mills theory and integrable systems. Nucl. Phys. B 460, 299–334 (1996). arXiv:hep-th/9510101
  10. 10.
    Dorey, N., Zhao, P.: Solution of quantum integrable systems from quiver gauge theories. arXiv:1512.09367
  11. 11.
    Etingof, P.: Calogero–Moser systems and representation theory. In: Zurich Lectures in Advanced Mathematics, EMS, Zurich (2007). arXiv:math/0606233
  12. 12.
    Fehér, L., Klimčík On the duality between the hyperbolic Sutherland and the rational Ruijsenaars–Schneider models. J. Phys. A 42, 185202, 13 (2009). arXiv:0901.1983
  13. 13.
    Fock, V., Gorsky, A., Nekrasov, N., Rubtsov, V.: Duality in integrable systems and gauge theories. J. High Energy Phys. (2000), Paper 28, 40. arXiv:hep-th/9906235
  14. 14.
    Gibbons, J., Hermsen, T.: A generalization of the Calogero–Moser system. Physica 11D, 337–348 (1984)ADSzbMATHGoogle Scholar
  15. 15.
    Ginzburg, V.: Non-commutative symplectic geometry, quiver varieties, and operads. Math. Res. Lett. 8, 377–400 (2001). arXiv:math/0005165
  16. 16.
    Ginzburg, V.: Lectures on noncommutative geometry. arXiv:math/0506603 (2005)
  17. 17.
    Gorsky, A., Nekrasov, N.: Relativistic Calogero–Moser model as gauged WZW theory. Nucl. Phys. B 436, 582–608 (1995). arXiv:hep-th/9401017
  18. 18.
    Ibort, A., Magri, F., Marmo, G.: Bihamiltonian structures and Stäckel separability. J. Geom. Phys. 33, 210–228 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Karoubi, M.: Homologie cyclique et \(K\)-théorie. Astérisque, p. 147 (1987)Google Scholar
  20. 20.
    Kontsevich, M.: Formal (non)commutative symplectic geometry. In: The Gel\(^{\prime }\)fand Mathematical Seminars, 1990–1992, pp. 173–187. Birkhäuser , Boston (1993)Google Scholar
  21. 21.
    Kosmann-Schwarzbach, Y., Magri, F.: Poisson–Nijenhuis structures. Annales de l’I.H.P. 53, 35–81 (1990)Google Scholar
  22. 22.
    Lazaroiu, C.I.: On the non-commutative geometry of topological D-branes. J. High Energy Phys. 2005, 032 (2005). arXiv:hep-th/0507222
  23. 23.
    Loday, J.-L.: Cyclic homology, vol. 301 Grundlehren der Mathematischen Wissenschaften, 2nd edn. Springer, Berlin (1998)Google Scholar
  24. 24.
    Magri, F.: A simple model of the integrable Hamiltonian equation. J. Math. Phys. 19, 1156–1162 (1978)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Magri, F., Marsico, T.: Some developments of the concept of Poisson manifold in the sense of A. Lichnerowicz. In: Ferrarese, G. (ed.) Gravitation, Electromagnetism, and Geometric Structures, pp. 207–222. Pitagora editrice, Bologna (1996)Google Scholar
  26. 26.
    Magri, F., Morosi, C.: A geometrical characterization of Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds. Quaderno S 19/1984, Università degli studi di Milano (1984)Google Scholar
  27. 27.
    Nekrasov, N.: Infinite-dimensional algebras, many-body systems and gauge theories. In: Moscow Seminar in Mathematical Physics, vol. 191 of Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, RI, pp. 263–299 (1999)Google Scholar
  28. 28.
    Odesskii, A., Rubtsov, V., Sokolov, V.: Double Poisson brackets on free associative algebras. In: Noncommutative birational geometry, representations and combinatorics, vol. 592 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2013, pp. 225–239. arXiv:1208.2935
  29. 29.
    Pichereau, A., Van de Weyer, G.: Double Poisson cohomology of path algebras of quivers. J. Algebra 319, 2166–2208 (2008). arXiv:math/0701837
  30. 30.
    Ruijsenaars, S.N.M.: Action-angle maps and scattering theory for some finite-dimensional integrable systems. I. The pure soliton case. Commun. Math. Phys. 115, 127–165 (1988)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Seiberg, N., Witten, E.: Electric-magnetic duality, monopole condensation, and confinement in \(N=2\) supersymmetric Yang-Mills theory. Nucl. Phys. B 426, 19–52 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Seiberg, N., Witten, E.: Monopoles, duality and chiral symmetry breaking in \(N=2\) supersymmetric QCD. Nucl. Phys. B 431, 484–550 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Tacchella, A.: An introduction to associative geometry with applications to integrable systems. J. Geom. Phys. (to appear). arXiv:1611.00644
  34. 34.
    Turiel, F.-J.: Structures bihamiltoniennes sur le fibré cotangent. C. R. Acad. Sci. Paris Sér. I Math. 315, 1085–1088 (1992)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Van den Bergh, M.: Double Poisson algebras. Trans. Am. Math. Soc. 360, 5711–5769 (2008). arXiv:math/0410528

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di GenovaGenoaItaly

Personalised recommendations