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On the Liouville Integrability of the Periodic Kostant–Toda Flow on Matrix Loops of Level k

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Abstract

In this work, we consider the periodic Kostant–Toda flow on matrix loops in \({\mathfrak{s}\mathfrak{l}(n,\mathbb{C})}\) of level k, which correspond to periodic infinite band matrices with period n with lower bandwidth equal to k and fixed upper bandwidth equal to 1 with 1’s on the first superdiagonal. We show that the coadjoint orbits through the submanifold of such matrix loops can be identified with those of a finite-dimensional Lie group, which appears in the form of a semi-direct product. We then characterize the generic coadjoint orbits and obtain an explicit global cross-section for such orbits. We also establish the Liouville integrability of the periodic Kostant–Toda flow on such orbits via the construction of action-angle variables.

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References

  1. Adler M.: On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-deVries type equations. Invent. Math. 50, 219–248 (1979)

    Article  ADS  MATH  Google Scholar 

  2. Adams M., Harnad J., Hurtubise J.: Isospectral Hamiltonian flows in finite and infinite dimensions. Commun. Math. Phys. 134, 555–585 (1990)

    Article  ADS  MATH  Google Scholar 

  3. Arnold J.M.: Soliton pulse-position modulation. IEEE Proc.-J. 140, 359–366 (1993)

    Google Scholar 

  4. Adler, M., van Moerbeke, P., Vanhaecke, P.: Algebraic integrability, Painleve geometry and Lie algebras. In: Ergebnisse der Mathematik. Springer, Berlin (2004)

  5. Aptekarev A.I.: Asymptotic properties of polynomials orthogonal on a system of contours and periodic motions of Toda lattices. Math. USSR Sbornik 53, 233–260 (1986)

    Article  MATH  Google Scholar 

  6. Baker H.F.: Examples of applications of Newton’s polygon to the theory of singular points of algebraic functions. Trans. Cambr. Philos. Soc. 15, 403–450 (1893)

    ADS  Google Scholar 

  7. Ben Abdeljelil K.: The integrability of the periodic full Kostant–Toda lattice on a simple Lie algebra. J. Lie Theory 21, 929–960 (2011)

    MathSciNet  MATH  Google Scholar 

  8. Babelon O., Bernard D., Talon M.: Introduction to Classical Integrable Systems. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  9. Benettin G., Christodoulidi H., Ponno A.: The Fermi–Pasta–Ulam problem and its underlying integrable dynamics. J. Stat. Phys. 152, 195–212 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  10. Brieskorn E.: Plane Algebraic Curves. Birkhäuser Verlag, Knörrer (1986)

    Book  MATH  Google Scholar 

  11. Beelen P., Pellikaan R.: The Newton polygon of plane curves with many rational points. Des. Codes Cryptogr. 21, 41–67 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Castryck, W., Cools, F.: On the Intrinsicness of the Newton Polygon (2013). arXiv:1304.4997

  13. Deift P., Li L.C., Nanda T., Tomei C.: The Toda flow on a generic orbit is integrable. Commun. Pure Appl. Math. 39, 183–232 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  14. Deift P., Li L.C., Tomei C.: Matrix factorizations and integrable systems. Commun. Pure Appl. Math. 42, 443–521 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  15. Date D., Tanaka S.: Analogue of inverse scattering theory for the discrete Hill’s equation and exact solutions for the periodic Toda lattice. Prog. Theor. Phys. 55, 457–465 (1976)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Ercolani, N., Flaschka, H., Singer, S.: The geometry of the full Kostant–Toda lattice. In: Integrable Systems (Luminy, 1991). Progr. Math., vol. 115, 181–225. Birkhäuser, Boston (1993)

  17. Fermi, E., Pasta, J., Ulam, S.: Studies of nonlinear problems. I. Technical Reprt LA 1940, Los Alamos Scientific Laboratory Report (1955)

  18. Flaschka H.: The Toda lattice. II. Existence of integrals. Phys. Rev. B 9, 1924–1925 (1974)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Flaschka H.: On the Toda lattice. II. Inverse-scattering solution. Prog. Theor. Phys. 51, 703–716 (1974)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Gao S.: Absolute irreducibility of polynomials via Newton polytopes. J. Alg. 237, 501–520 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gekhtman M., Shapiro M.: Noncommutative and commutative integrability of generic Toda flows in simple Lie algebras. Commun. Pure Appl. Math. 52, 53–84 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Givental A., Kim B.: Quantum cohomology of flag manifolds and Toda lattices. Commun. Math. Phys. 168, 609–641 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Griffiths P.: Linearizing flows and a cohomological interpretation of Lax equations. Am. J. Math. 107, 1445–1484 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  24. Guest M.: Harmonic Maps, Loop Groups, and Integrable Systems. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  25. Joseph A.: An algebraic slice in the coadjoint space of the Borel and the Coxeter element. Adv. Math. 227, 522–585 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Khovanskii A.G.: Newton polyhedra and the genus of complete intersections. Funct. Anal. Appl. 12, 38–46 (1978)

    Article  MathSciNet  Google Scholar 

  27. Kostant B.: The solution to a generalized Toda lattice and representation theory. Adv. Math. 34, 195–338 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kostant B.: Lie group representations on polynomial rings. Am. J. Math. 85, 327–404 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  29. Knuth, D.: The art of computer programming. In: Fundamental Algorithms, vol. 1, p. 42. Addison-Wesley, Reading (1973)

  30. Li, L.-C., Nie, Z.: Integrability of the periodic Kostant–Toda flow on matrix loops of level k. C. R. Math. Acad. Sci. Paris 353, 363–367 (2015)

  31. Manakov S.V.: Complete integrability and stochastization of discrete dynamical systems. Sov. Phys. JETP 40, 269–274 (1975)

    ADS  MathSciNet  Google Scholar 

  32. Mikhailov A.V., Olshanetsky M.A., Perelomov A.M.: Two dimensional generalized Toda lattice. Commun. Math. Phys. 79, 473–488 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  33. van Moerbeke P., Mumford D.: The spectrum of difference operators and algebraic curves. Acta Math. 143, 93–154 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  34. Moser, J.: Finitely many mass points on the line under the influence of an exponential potential-an integrable system. Dynamical systems, theory and applications. Lecture Notes in Physics, vol. 38, pp. 467–497. Springer, Berlin (1975)

  35. Noether M.: Rationale Ausführungen der Operationen in der Theorie der algebraischen Functionen. Math. Ann. 23, 311–358 (1884)

    Article  MathSciNet  Google Scholar 

  36. Olive D.I., Turok N.: The symmetries of Dynkin diagrams and reduction of Toda field equations. Nucl. Phys. B 215, 470–494 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  37. Olive D.I., Turok N.: Algebraic structure of Toda systems. Nucl. Phys. B 220, 491–507 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  38. Semenov-Tian-Shansky M.: What is a classical r-matrix?. Funct. Anal. Appl. 17, 259–272 (1983)

    Article  MATH  Google Scholar 

  39. Symes W.: The QR algorithm and scattering for the finite nonperiodic Toda lattice. Physica D 4, 275–280 (1982)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Toda M.: Vibration of a chain with nonlinear interaction. J. Phys. Soc. Japan 22, 431–436 (1967)

    Article  ADS  Google Scholar 

  41. Tauvel P., Yu R.W.T.: Affine slice for the coadjoint action of a class of biparabolic subalgebras of a semisimple Lie algebra. Algebr. Represent. Theory 16, 859–872 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ueno, K., Takasaki, K.: Toda lattice hierarchy I. Proc. Japan Acad. Ser. A 59, 167–170 (1983)

  43. Walker R.: Algebraic Curves. Springer, New York (1991)

    Google Scholar 

  44. Wilson G.: The modified Lax and two-dimensional Toda lattice equations associated with simple Lie algebras. Ergodic Theory Dynam. Syst. 1, 361–380 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zakharov V.E., Musher S.L., Rubenchik A.M.: Weak Langmuir turbulence of an isothermal plasma. Sov. Phys. JETP 42, 80–86 (1975)

    ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Luen-Chau Li

  2. Department of Mathematics and Statistics, Utah State University, Logan, UT, 84322-3900, USA

    Zhaohu Nie

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  1. Luen-Chau Li
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  2. Zhaohu Nie
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Correspondence to Luen-Chau Li.

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Communicated by P. Deift

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Li, LC., Nie, Z. On the Liouville Integrability of the Periodic Kostant–Toda Flow on Matrix Loops of Level k . Commun. Math. Phys. 352, 1153–1203 (2017). https://doi.org/10.1007/s00220-016-2768-7

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  • DOI: https://doi.org/10.1007/s00220-016-2768-7

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