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Theoretical and Mathematical Physics

, Volume 167, Issue 3, pp 772–784 | Cite as

Variational Lie algebroids and homological evolutionary vector fields

  • A. V. KiselevEmail author
  • J. W. van de Leur
Article

Abstract

We define Lie algebroids over infinite jet spaces and obtain their equivalent representation in terms of homological evolutionary vector fields.

Keywords

Lie algebroid BRST differential Poisson structure integrable system string theory 

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References

  1. 1.
    A. Yu. Vaintrob, Russ. Math. Surveys, 52, 428–429 (1997).ADSzbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    M. Alexandrov, A. Schwarz, O. Zaboronsky, and M. Kontsevich, Internat. J. Mod. Phys. A, 12, 1405–1429 (1997); arXiv:hep-th/9502010v2 (1995).ADSzbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Y. Kosmann-Schwarzbach, SIGMA, 0804, 005 (2008); arXiv:0710.3098v3 [math.SG] (2007); Lett. Math. Phys., 69, 61–87 (2004); arXiv:math/0312524 (2003); Ann. Inst. Fourier (Grenoble), 46, 1243–1274 (1996).MathSciNetGoogle Scholar
  4. 4.
    A. V. Bocharov et al., Symmetries and Conservation Laws for Equations of Mathematical Physics [in Russian], Faktorial, Moscow (2005); English transl. prev. ed.: Symmetries and Conservation Laws for Differential Equations of Mathematical Physics (Trans. Math. Monogr., Vol. 182), Amer. Math. Soc., Providence, R. I. (1999).Google Scholar
  5. 5.
    T. Voronov, “Graded manifolds and Drinfeld doubles for Lie bialgebroids,” in: Quantization, Poisson Brackets, and Beyond (Contemp. Math., Vol. 315, T. Voronov, ed.), Amer. Math. Soc., Providence, R. I. (2002), pp. 131–168.CrossRefGoogle Scholar
  6. 6.
    J.-C. Herz, C. R. Acad. Sci. Paris, 263,1935–1937, 2289–2291 (1953).MathSciNetGoogle Scholar
  7. 7.
    Y. Kosmann-Schwarzbach and F. Magri, Ann. Inst. H. Poincaré, 53, 35–81 (1990).zbMATHMathSciNetGoogle Scholar
  8. 8.
    A. V. Kiselev and J. W. van de Leur, Theor. Math. Phys., 162, 149–162 (2010); arXiv:0902.3624v2 [nlin.SI] (2009).zbMATHCrossRefGoogle Scholar
  9. 9.
    A. N. Leznov and M. V. Saveliev, Lett. Math. Phys., 3, 489–494 (1979).ADSzbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    A. V. Zhiber and V. V. Sokolov, Russ. Math. Surveys, 56, 61–101 (2001).ADSzbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    A. Kumpera and D. C. Spencer, Lie Equations (Ann. Math. Stud., Vol. 73), Vol. 1, General theory, Princeton Univ. Press, Princeton, N. J. (1972).zbMATHGoogle Scholar
  12. 12.
    J. Krasil’shchik and A. Verbovetsky, “Geometry of jet spaces and integrable systems,” J. Geom. Phys. (to appear); arXiv:1002.0077v5 [math.DG] (2010).Google Scholar
  13. 13.
    B. A. Kupershmidt, “Geometry of jet bundles and the structure of Lagrangian and Hamiltonian formalisms,” in: Geometric Methods in Mathematical Physics (Lect. Notes Math., Vol. 775), Springer, Berlin (1980), pp. 162–218.CrossRefGoogle Scholar
  14. 14.
    P. J. Olver, Applications of Lie Groups to Differential Equations (Grad. Texts Math., Vol. 107), Springer, New York (1993).zbMATHCrossRefGoogle Scholar
  15. 15.
    A. V. Kiselev, Theor. Math. Phys., 144, 952–960 (2005); arXiv:nlin.SI/0409061v2 (2004).zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    A. V. Kiselev and J. W. van de Leur, J. Phys. A, 42, 404011 (2009); arXiv:0903.1214v1 [nlin.SI] (2009).CrossRefMathSciNetGoogle Scholar
  17. 17.
    A. V. Kiselev and J. W. van de Leur, “Involutive distributions of operator-valued evolutionary vector fields and their affine geometry,” Preprint No. IHES/M/07/38, Inst. Hautes Études Sci., Bures-sur-Yvette (2007).Google Scholar
  18. 18.
    P. Kersten, I. Krasil’shchik, and A. Verbovetsky, J. Geom. Phys., 50, 273–302 (2004); arXiv:math/0304245v5 (2003).ADSzbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    M. Nesterenko and R. Popovych, J. Math. Phys., 47, 123515 (2006); arXiv:math-ph/0608018v4 (2006).ADSCrossRefMathSciNetGoogle Scholar
  20. 20.
    C. P. Boyer and J. D. Finley III, J. Math. Phys., 23, 1126–1130 (1982).ADSzbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    V. A. Golovko, I. S. Krasil’shchik, and A. M. Verbovetsky, Theor. Math. Phys., 154, 227–239 (2008); arXiv:0812.4684v3 [math.DG] (2008).zbMATHCrossRefGoogle Scholar
  22. 22.
    T. Lada and J. Stasheff, Internat. J. Theoret. Phys., 32, 1087–1103 (1993); arXiv:hep-th/9209099v1 (1992).ADSzbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    A. V. Kiselev, J. Math. Sci., 141, 1016–1030 (2007); arXiv:math.RA/0410185v1 (2004).CrossRefMathSciNetGoogle Scholar
  24. 24.
    P. Kersten, I. Krasil’shchik, and A. Verbovetsky, Acta Appl. Math., 83, 167–173 (2004); arXiv:math/0310451v2 (2003).zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    P. A. M. Dirac, Lectures on Quantum Mechanics, Acad. Press, New York (1967); T. de Donder, Théorie invariantive du calcul des variations, Gauthier-Villars, Paris (1930).Google Scholar
  26. 26.
    J. Krasil’shchik and A. Verbovetsky, “Homological methods in equations of mathematical physics,” arXiv: math.DG/9808130v2 (1998).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of UtrechtUtrechtThe Netherlands
  2. 2.Johann Bernoulli Institute for Mathematics and Computer ScienceUniversity of GroningenGroningenThe Netherlands

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