Poisson–Lie Algebras and Singular Symplectic Forms Associated to Corank 1 Type Singularities

Abstract

We show that there exists a natural Poisson–Lie algebra associated to a singular symplectic structure \(\omega\). We construct Poisson–Lie algebras for the Martinet and Roussarie types of singularities. In the special case when the singular symplectic structure is given by the pullback from the Darboux form, \(\omega=F^*\omega_0\), this Poisson–Lie algebra is a basic symplectic invariant of the singularity of the smooth mapping \(F\) into the symplectic space \(({\mathbb R}^{2n},\omega_0)\). The case of \(A_k\) singularities of pullbacks is considered, and Poisson–Lie algebras for \(\Sigma_{2,0}\), \(\Sigma_{2,2,0}^{\textrm{e}}\) and \(\Sigma_{2,2,0}^{\textrm{h}}\) stable singularities of \(2\)-forms are calculated.

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References

  1. 1

    V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps, Vol. 1: The Classification of Critical Points, Caustics and Wave Fronts (Birkhäuser, Boston, 1985), Monogr. Math. 82.

    Book  Google Scholar 

  2. 2

    J. W. Bruce and P. J. Giblin, Curves and Singularities (Cambridge Univ. Press, Cambridge, 1992).

    Book  Google Scholar 

  3. 3

    A. A. Davydov, “Normal form of a differential equation, not solvable for the derivative, in a neighborhood of a singular point,” Funct. Anal. Appl. 19 (2), 81–89 (1985) [transl. from Funkts. Anal. Prilozh. 19 (2), 1–10 (1985)].

    Article  Google Scholar 

  4. 4

    A. A. Davydov and J. Basto-Gonçalves, “Controllability of generic inequalities near singular points,” J. Dyn. Control Syst. 7 (1), 77–99 (2001).

    MathSciNet  Article  Google Scholar 

  5. 5

    T. Fukuda, “Local topological properties of differentiable mappings. I,” Invent. Math. 65, 227–250 (1981).

    MathSciNet  Article  Google Scholar 

  6. 6

    T. Fukuda and S. Janeczko, “Singularities of implicit differential systems and their integrability,” in Geometric Singularity Theory (Pol. Acad. Sci., Inst. Math., Warsaw, 2004), Banach Cent. Publ. 65, pp. 23–47.

    MathSciNet  Article  Google Scholar 

  7. 7

    T. Fukuda and S. Janeczko, “Global properties of integrable implicit Hamiltonian systems,” in Singularity Theory: Proc. 2005 Marseille Singularity Sch. Conf. (World Scientific, Singapore, 2007), pp. 593–611.

    MathSciNet  Article  Google Scholar 

  8. 8

    T. Fukuda and S. Janeczko, “On the Poisson algebra of a singular map,” J. Geom. Phys. 86, 194–202 (2014).

    MathSciNet  Article  Google Scholar 

  9. 9

    T. Fukuda and S. Janeczko, “Symplectic singularities and solvable Hamiltonian mappings,” Demonstr. Math. 48 (2), 118–146 (2015).

    MathSciNet  MATH  Google Scholar 

  10. 10

    M. Golubitsky and D. Tischler, “An example of moduli for singular symplectic forms,” Invent. Math. 38, 219–225 (1977).

    MathSciNet  Article  Google Scholar 

  11. 11

    H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics (Birkhäuser, Basel, 1994), Birkhäuser Adv. Texts.

    Book  Google Scholar 

  12. 12

    G. Ishikawa, “Symplectic and Lagrange stabilities of open Whitney umbrellas,” Invent. Math. 126 (2), 215–234 (1996).

    MathSciNet  Article  Google Scholar 

  13. 13

    G. Ishikawa, “Determinacy, transversality and Lagrange stability,” in Geometry and Topology of Caustics—CAUSTICS’98: Proc. Banach Cent. Symp., Warsaw, 1998 (Pol. Acad. Sci., Inst. Math., Warsaw, 1999), Banach Cent. Publ. 50, pp. 123–135.

    MathSciNet  MATH  Google Scholar 

  14. 14

    S. Janeczko, “On implicit Lagrangian differential systems,” Ann. Pol. Math. 74, 133–141 (2000).

    MathSciNet  Article  Google Scholar 

  15. 15

    J. Martinet, “Sur les singularités des formes différentielles,” Ann. Inst. Fourier 20 (1), 95–178 (1970).

    MathSciNet  Article  Google Scholar 

  16. 16

    J. Martinet, “Miscellaneous results and problems about differential forms and differential systems,” in Global Analysis and Applications: Int. Sem. Course, Trieste, 1972 (Int. At. Energy Agency, Vienna, 1974), Vol. 3, pp. 41–46.

    MathSciNet  MATH  Google Scholar 

  17. 17

    B. Morin, “Formes canoniques des singularités d’une application différentiable,” C. R. Acad. Sci. Paris 260, 5662–5665, 6503–6506 (1965).

    MathSciNet  MATH  Google Scholar 

  18. 18

    R. Roussarie, Modèles locaux de champs et de formes (Soc. Math. France, Paris, 1976), Astérisque 30.

    MATH  Google Scholar 

  19. 19

    F. Takens, “Implicit differential equations: Some open problems,” in Singularités d’applications différentiables (Springer, Berlin, 1976), Lect. Notes Math. 535, pp. 237–253.

    MathSciNet  Article  Google Scholar 

  20. 20

    G. Wassermann, Stability of Unfoldings (Springer, Berlin, 1974), Lect. Notes Math. 393.

    Book  Google Scholar 

  21. 21

    A. Weinstein, Lectures on Symplectic Manifolds (Am. Math. Soc., Providence, RI, 1977), CBMS Reg. Conf. Ser. Math. 29.

    Book  Google Scholar 

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Acknowledgments

The authors are grateful to the referees for helpful suggestions.

Funding

The work was partially supported by the NCN grant no. DEC-2013/11/B/ST1/03080.

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Correspondence to T. Fukuda.

Additional information

Dedicated to Professor Armen Sergeev on his 70th birthday

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Fukuda, T., Janeczko, S. Poisson–Lie Algebras and Singular Symplectic Forms Associated to Corank 1 Type Singularities. Proc. Steklov Inst. Math. 311, 129–151 (2020). https://doi.org/10.1134/S0081543820060085

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Keywords

  • implicit Hamiltonian system
  • solvability
  • singularities
  • Poisson–Lie algebra
  • singular symplectic structures