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Theoretical and Mathematical Physics

, Volume 198, Issue 2, pp 271–283 | Cite as

Two Problems in the Theory Of Differential Equations

  • D. A. LeitesEmail author
Article
  • 16 Downloads

Abstract

Differential equations considered in terms of exterior differential forms, as did É. Cartan, distinguish a differential ideal in the supercommutative superalgebra of differential forms, i.e., an affine supervariety. Therefore, each differential equation has a supersymmetry (perhaps trivial). Which superymmetries of systems of classical differential equations are not yet found? We also consider the question of why criteria of the formal integrability of differential equations are currently never used in practice.

Keywords

supersymmetry nonholonomic mechanics 

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References

  1. 1.
    K. Efetov, Supersymmetry in Disorder and Chaos, Cambridge Univ. Press, Cambridge (1997).zbMATHGoogle Scholar
  2. 2.
    D. Leites, “New Lie superalgebras, and mechanics,” Soviet Math. Dokl., 18, 1277–1280 (1977).zbMATHGoogle Scholar
  3. 3.
    V. G. Kac, “Lie superalgebras,” Adv. Math., 26, 8–96 (1977).CrossRefzbMATHGoogle Scholar
  4. 4.
    P. Grozman, “Classification of bilinear invariant operators on tensor fields,” Funct. Anal. Appl., 14, 127–128 (1980); arXiv:math/0509562v1 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    D. Leites and I. Shchepochkina, “Classification of simple Lie superalgebras of vector fields,” Preprint MPIMBonn 2003–28, http://www.mpim–bonn.mpg.de/preprints/, Max Planck Inst. Math., Bonn (2003).zbMATHGoogle Scholar
  6. 6.
    V. Kac, “Classification of infinite–dimensional simple linearly compact Lie superalgebras,” Adv. Math., 139, 1–55 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    I. Shchepochkina, “Five exceptional simple Lie superalgebras of vector fields and their fourteen regradings,” Represent. Theory, 3, 373–415 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    N. Cantarini and V. G. Kac, “Infinite–dimensional primitive linearly compact Lie superalgebras,” Adv. Math., 207, 328–419 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    N. Cantarini and V. G. Kac, “Classification of linearly compact simple rigid superalgebras,” IMRN, 2010, 3341–3393 (2010); arXiv:0909.3100v1 [math.QA] (2009).MathSciNetzbMATHGoogle Scholar
  10. 10.
    S.–J. Cheng and V. Kac, “Generalized Spencer cohomology and filtered deformations of Z–graded Lie superalgebras,” Adv. Theor. Math. Phys., 2, 1141–1182 (1998); Addendum, 8, 697–709 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    S. J. Cheng and V. G. Kac, “Structure of some Z–graded Lie superalgebras of vector fields,” Transform. Groups, 4, 219–272 (1999); Erratum, 9, 399–400 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    D. Leites and I. Shchepochkina, “How should the antibracket be quantized?” Theor. Math. Phys., 126, 281–306 (2001); arXiv:math–ph/0510048v1 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Sh.–J. Cheng and V. G. Kac, “Generalized Spencer cohomology and filtered deformations of Z–graded Lie superalgebras,” Adv. Theor. Math. Phys., 2, 1141–1182 (1998); arXiv:math.RT/9805039v3 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    B. Kostant, Graded Manifolds, Graded Lie Groups, and Prequantisation (Lect. Notes Math., Vol. 570), Springer, Berlin (1975).Google Scholar
  15. 15.
    I. A. Batalin and G. A. Vilkovisky, “Gauge algebra and quantization,” Phys. Lett. B, 102, 27–31 (1981); “Quantization of gauge theories with linearly dependent generators,” Phys. Rev. D, 28, 2567–2582 (1983); Erratum, 30, 508 (1984).ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    V. Dotsenko, S. Shadrin, and B. Vallette, “Givental group action on topological field theories and homotopy Batalin–Vilkovisky algebras,” Adv. Math., 236, 224–256 (2013); arXiv:1112.1432v5 [math.QA] (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    P. Grozman, “SuperLie: A Mathematica package for calculations in Lie algebras and superalgebras,” http://www.equaonline.com/math/SuperLie (2013).Google Scholar
  18. 18.
    R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths, Exterior Differential Systems (Math. Sci. Res. Inst. Publ., Vol. 18), Springer, New York (1991).CrossRefzbMATHGoogle Scholar
  19. 19.
    P. Deligne, P. Etingof, D. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D. Morrison, and E. Witten, eds., Quantum Fields and Strings: A Course for Mathematicians, Vol. 1, Amer. Math. Soc., Providence, R. I. (1999).zbMATHGoogle Scholar
  20. 20.
    J. Bernstein, D. Leites, V. Molotkov, and V. Shander, Seminar on Supermanifolds [in Russian], Vol. 1, Algebra and Calculus: Main Chapters, MCCME, Moscow (2011).Google Scholar
  21. 21.
    F. Berezin, Introduction to Superanalysis [in Russian] (D. Leites, ed. With appendices by D. Leites, V. Shander, and I. Shchepochkina), MCCME, Moscow (2013).Google Scholar
  22. 22.
    D. Leites, “The Riemann tensor for nonholonomic manifolds,” Homology Homotopy Appl., 4, 397–407 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    I. Shchepochkina, “How to realize Lie algebras by vector fields,” Theor. Math. Phys., 147, 821–838 (2006); arXiv:math.RT/0509472v1 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    V. N. Shander, “Complete integrability of ordinary differential equations on supermanifolds,” Funct. Anal. Appl., 17, 74–75 (1983).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    V. Molotkov, “Infinite–dimensional and colored supermanifolds,” J. Nonlinear Math. Phys., 17, suppl. 1, 375–446 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    D. Giulini, “The superspace of geometrodynamics,” Gen. Rel. Grav., 41, 785–815 (2009); arXiv:0902.3923v1 [gr–qc] (2009).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    D. A. Leites, “Spectra of graded–commutative rings,” Russian Math. Surveys, 29, 209–210 (1974).MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    W. I. Fushchich and A. G. Nikitin, Symmetries of Maxwell’s Equations (Math. Its Appl., Vol. 8), Reidel, Dordrecht (1987)CrossRefzbMATHGoogle Scholar
  29. 28a.
    J. Niederle and A. G. Nikitin, “Extended supersymmetries for the Schrödinger–Pauli equation,” J. Math. Phys., 40, 1280–1293 (1999).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 29.
    S. Duplij, J. Bagger, and W. Siegel, eds., Concise Encyclopedia of Supersymmetry: And Noncommutative Structures in Mathematics and Physics, Kluwer, Dordrecht (2003).zbMATHGoogle Scholar
  31. 30.
    S. Bouarroudj, P. Grozman, S. Leites, and I. Shchepochkina, “Minkowski superspaces and superstrings as almost real–complex supermanifolds,” Theor. Math. Phys., 173, 1687–1708 (2012); arXiv:1010.4480v2 [math.DG] (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  32. 31.
    É. Cartan, OEuvres complètes. Partie II: Algèbre, systèmes différentiels et problèmes d’équivalence, CNRS, Paris (1984).Google Scholar
  33. 32.
    J. A. Wheeler, Einstein’s Vision, Springer, Berlin (1968).CrossRefGoogle Scholar
  34. 33.
    Yu. I. Manin, Gauge Fields and Complex Geometry [in Russian], Nauka, Moscow (1984); English transl.: Gauge Field Theory and Complex Geometry (Grundlehren Math. Wiss., Vol. 289), Springer, Berlin (1997).Google Scholar
  35. 34.
    A. V. Zorich, “Integration of pseudodifferential forms and inversion of Radon–type integral transformations,” Russian Math. Surveys, 42, 151–152 (1987).ADSCrossRefzbMATHGoogle Scholar
  36. 35.
    D. Quillen, “Superconnections and the Chern character, topology,” Internat. J. Math., 24, 89–95 (1985).MathSciNetzbMATHGoogle Scholar
  37. 36.
    N. Berline, E. Getzler, and M. Vergne, Heat Kernels and Dirac Operators (Grundlehren Math. Wiss., Vol. 298), Springer, Berlin (1992).CrossRefzbMATHGoogle Scholar
  38. 37.
    D. Leites, “The index theorem for homogeneous differential operators on supermanifolds,” in: Supersymmetries and Quantum Symmetries SQS’ 99 (E. Ivanov, S. Krivonos, and A. Pashnev, eds.), Joint Inst. Nucl. Res., Dubna (2000), pp. 405–408; arXiv:math–ph/0202024v1 (2002).Google Scholar
  39. 38.
    D. Leites and I. Shchepochkina, “The Howe duality and Lie superalgebras,” in: Noncommutative Structures in Mathematics and Physics (NATO Sci. Ser. II Math. Phys. Chem., Vol. 22, S. Duplij and J. Wess, eds.), Springer, Dordrecht (2001), pp. 93–111; arXiv:math.RT/0202181v1 (2002).Google Scholar
  40. 39.
    J. Wess and J. Bagger, Supersymmetry and Supergravity, Princeton Univ. Press, Princeton, N. J. (1992).zbMATHGoogle Scholar
  41. 40.
    E. Witten, “An interpretation of classical Yang–Mills theory,” Phys. Lett. B, 77, 394–398 (1978).ADSCrossRefGoogle Scholar
  42. 41.
    E. Witten, “Dynamical breaking of supersymmetry,” Nucl. Phys. B, 188, 513–554 (1981).ADSCrossRefzbMATHGoogle Scholar
  43. 42.
    G. K. Gendenshtein and I. V. Krive, “Supersymmetry in quantum mechanics,” Sov. Phys. Usp., 28, 645–666 (1985).ADSMathSciNetCrossRefGoogle Scholar
  44. 43.
    J. Brundan, “Kazhdan–Lusztig polynomials and character formulae for the Lie superalgebra q(n),” Adv. Math., 182, 28–77 (2004); arXiv:math/0207024v2 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  45. 44.
    S.–J. Cheng and J.–H. Kwon, “Finite–dimensional half–integer weight modules over queer Lie superalgebras,” Commun. Math. Phys., 346, 945–965 (2016); arXiv:1505.06602v1 [math.RT] (2015).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. 45.
    A. A. Kirillov, “Orbits of the group of diffeomorphisms of the circle and local Lie superalgebras,” Funct. Anal. Appl., 15, 135–137 (1981).MathSciNetCrossRefzbMATHGoogle Scholar
  47. 46.
    P. Grozman, D. Leites, and I. Shchepochkina, “Lie superalgebras of string theories,” Acta Math. Vietnam., 26, 27–63 (2001); arXiv:hep–th/9702120v1 (1997).MathSciNetzbMATHGoogle Scholar
  48. 47.
    V. Yu. Ovsienko, O. Ovsienko, and Yu. Chekanov, “Classification of contact–projective structures on supercircles,” Russian Math. Surveys, 44, 212–213 (1989).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. 48.
    D. Leites, “Supersymmetry of the Sturm–Liouville and Korteveg–de Vries operators,” in: Operator Methods in Ordinary and Partial Differential Equations (Oper. Theor. Adv. Appl., Vol. 132, S. Albeverio, N. Elander, W. N. Everitt, and P. Kurasov, eds.), Birkhäuser, Basel (2002), pp. 267–285.Google Scholar
  50. 49.
    S. Mohammadzadeh and D. B. Fuchs, “Cohomology of the Lie algebra H2: Experimental results and conjectures,” Funct. Anal. Appl., 48, 128–137 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  51. 50.
    V. Kornyak, “Computation of cohomology of Lie superalgebras of vector fields,” Internat. J. Modern Phys. C, 11, 397–413 (2000); arXiv:math/0002210v1 (2000).ADSMathSciNetGoogle Scholar
  52. 51.
    P. Grozman and D. Leites, “From supergravity to ballbearings,” in: Supersymmetries and Quantum Symmetries SQS’ 97 (Lect. Notes Phys., Vol. 524, J. Wess and E. Ivanov, eds.), Springer, Berlin (1999), pp. 58–67.Google Scholar
  53. 52.
    A. Galperin, E. Ivanov, V. Ogievetsky, and E. Sokatchev, Harmonic Superspace, Cambridge Univ. Press, Cambridge (2001).CrossRefzbMATHGoogle Scholar
  54. 53.
    I. Zelenko, “On Tanaka’s prolongation procedure for filtered structures of constant type,” SIGMA, 5, 094 (2009).MathSciNetzbMATHGoogle Scholar
  55. 54.
    D. Alekseevsky and L. David, “Prolongation of Tanaka structures: An alternative approach,” Ann. Mat., 196, 1137–1164 (2017); arXiv:1603.00700v2 [math.DG] (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  56. 55.
    D. The, “Exceptionally simple PDE,” Differ. Geom. Appl., 56, 13–41 (2018); arXiv:1603.08251v2 [math.DG] (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  57. 56.
    V. M. Sergeev, The Limits of Rationality: A Thermodynamical Approach to Market Economy [in Russian], Fasis, Moscow (1999).Google Scholar
  58. 57.
    V. P. Pavlov, Dirac’s Nonholonomic Mechanics and Differential Geometry (Lecture Courses, Vol. 22), MIAN, Moscow (2014).Google Scholar
  59. 58.
    V. P. Pavlov and V. M. Sergeev, “Thermodynamics from the differential geometry standpoint,” Theor. Math. Phys., 157, 1484–1490 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  60. 59.
    V. Sergeev, “The thermodynamic approach to markets,” arXiv:0803.3432v1 [physics.soc–ph] (2008).Google Scholar
  61. 60.
    E. Poletaeva, “The analogs of Riemann and Penrose tensors on supermanifolds,” arXiv:math/0510165v1 [math.RT] (2005).Google Scholar
  62. 61.
    D. B. Fuks, Cohomology of Infinite–Dimensional Lie Algebras, Consultants Bureau, New York (1986).CrossRefzbMATHGoogle Scholar
  63. 62.
    B. Feigin and D. Fuchs, “Cohomologies of Lie groups and Lie algebras [in Russian],” in: Sovrem. Probl. Mat. Fund. Naprav., Vol. 21, VINITI, Moscow (1988), pp. 121–209; English transl. in: Lie Groups and Lie Algebras: II. Discrete Subgroups of Lie Groups and Cohomologies of Lie Groups and Lie Algebras (Encycl. Math. Sci., Vol. 21), Springer, Berlin (2000), pp. 125–223.zbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Division of Science and MathematicsNew York University Abu DhabiAbu DhabiUnited Arab Emirates
  2. 2.Department of MathematicsStockholm UniversityStockholmSweden

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