Communications in Mathematical Physics

, Volume 352, Issue 3, pp 1153–1203 | Cite as

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

  • Luen-Chau LiEmail author
  • Zhaohu Nie


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. A.
    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)ADSCrossRefzbMATHGoogle Scholar
  2. AHH.
    Adams M., Harnad J., Hurtubise J.: Isospectral Hamiltonian flows in finite and infinite dimensions. Commun. Math. Phys. 134, 555–585 (1990)ADSCrossRefzbMATHGoogle Scholar
  3. Ar.
    Arnold J.M.: Soliton pulse-position modulation. IEEE Proc.-J. 140, 359–366 (1993)Google Scholar
  4. AvMV.
    Adler, M., van Moerbeke, P., Vanhaecke, P.: Algebraic integrability, Painleve geometry and Lie algebras. In: Ergebnisse der Mathematik. Springer, Berlin (2004)Google Scholar
  5. Ap.
    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)CrossRefzbMATHGoogle Scholar
  6. B.
    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)ADSGoogle Scholar
  7. BA.
    Ben Abdeljelil K.: The integrability of the periodic full Kostant–Toda lattice on a simple Lie algebra. J. Lie Theory 21, 929–960 (2011)MathSciNetzbMATHGoogle Scholar
  8. BBT.
    Babelon O., Bernard D., Talon M.: Introduction to Classical Integrable Systems. Cambridge University Press, Cambridge (2003)CrossRefzbMATHGoogle Scholar
  9. BCP.
    Benettin G., Christodoulidi H., Ponno A.: The Fermi–Pasta–Ulam problem and its underlying integrable dynamics. J. Stat. Phys. 152, 195–212 (2013)ADSMathSciNetCrossRefGoogle Scholar
  10. BK.
    Brieskorn E.: Plane Algebraic Curves. Birkhäuser Verlag, Knörrer (1986)CrossRefzbMATHGoogle Scholar
  11. BP.
    Beelen P., Pellikaan R.: The Newton polygon of plane curves with many rational points. Des. Codes Cryptogr. 21, 41–67 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. CC.
    Castryck, W., Cools, F.: On the Intrinsicness of the Newton Polygon (2013). arXiv:1304.4997
  13. DLNT.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. DLT.
    Deift P., Li L.C., Tomei C.: Matrix factorizations and integrable systems. Commun. Pure Appl. Math. 42, 443–521 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  15. DT.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. EFS.
    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)Google Scholar
  17. FPU.
    Fermi, E., Pasta, J., Ulam, S.: Studies of nonlinear problems. I. Technical Reprt LA 1940, Los Alamos Scientific Laboratory Report (1955)Google Scholar
  18. F1.
    Flaschka H.: The Toda lattice. II. Existence of integrals. Phys. Rev. B 9, 1924–1925 (1974)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. F2.
    Flaschka H.: On the Toda lattice. II. Inverse-scattering solution. Prog. Theor. Phys. 51, 703–716 (1974)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. G.
    Gao S.: Absolute irreducibility of polynomials via Newton polytopes. J. Alg. 237, 501–520 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  21. GS.
    Gekhtman M., Shapiro M.: Noncommutative and commutative integrability of generic Toda flows in simple Lie algebras. Commun. Pure Appl. Math. 52, 53–84 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  22. GK.
    Givental A., Kim B.: Quantum cohomology of flag manifolds and Toda lattices. Commun. Math. Phys. 168, 609–641 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. Gr.
    Griffiths P.: Linearizing flows and a cohomological interpretation of Lax equations. Am. J. Math. 107, 1445–1484 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Gu.
    Guest M.: Harmonic Maps, Loop Groups, and Integrable Systems. Cambridge University Press, Cambridge (1997)CrossRefzbMATHGoogle Scholar
  25. J.
    Joseph A.: An algebraic slice in the coadjoint space of the Borel and the Coxeter element. Adv. Math. 227, 522–585 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. Kho.
    Khovanskii A.G.: Newton polyhedra and the genus of complete intersections. Funct. Anal. Appl. 12, 38–46 (1978)MathSciNetCrossRefGoogle Scholar
  27. K1.
    Kostant B.: The solution to a generalized Toda lattice and representation theory. Adv. Math. 34, 195–338 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  28. K2.
    Kostant B.: Lie group representations on polynomial rings. Am. J. Math. 85, 327–404 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  29. Kn.
    Knuth, D.: The art of computer programming. In: Fundamental Algorithms, vol. 1, p. 42. Addison-Wesley, Reading (1973)Google Scholar
  30. LN.
    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)Google Scholar
  31. M.
    Manakov S.V.: Complete integrability and stochastization of discrete dynamical systems. Sov. Phys. JETP 40, 269–274 (1975)ADSMathSciNetGoogle Scholar
  32. MOP.
    Mikhailov A.V., Olshanetsky M.A., Perelomov A.M.: Two dimensional generalized Toda lattice. Commun. Math. Phys. 79, 473–488 (1981)ADSMathSciNetCrossRefGoogle Scholar
  33. vMM.
    van Moerbeke P., Mumford D.: The spectrum of difference operators and algebraic curves. Acta Math. 143, 93–154 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  34. Mos.
    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)Google Scholar
  35. N.
    Noether M.: Rationale Ausführungen der Operationen in der Theorie der algebraischen Functionen. Math. Ann. 23, 311–358 (1884)MathSciNetCrossRefGoogle Scholar
  36. OT1.
    Olive D.I., Turok N.: The symmetries of Dynkin diagrams and reduction of Toda field equations. Nucl. Phys. B 215, 470–494 (1983)ADSMathSciNetCrossRefGoogle Scholar
  37. OT2.
    Olive D.I., Turok N.: Algebraic structure of Toda systems. Nucl. Phys. B 220, 491–507 (1983)ADSMathSciNetCrossRefGoogle Scholar
  38. STS.
    Semenov-Tian-Shansky M.: What is a classical r-matrix?. Funct. Anal. Appl. 17, 259–272 (1983)CrossRefzbMATHGoogle Scholar
  39. Sy.
    Symes W.: The QR algorithm and scattering for the finite nonperiodic Toda lattice. Physica D 4, 275–280 (1982)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. T.
    Toda M.: Vibration of a chain with nonlinear interaction. J. Phys. Soc. Japan 22, 431–436 (1967)ADSCrossRefGoogle Scholar
  41. TY.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  42. UT.
    Ueno, K., Takasaki, K.: Toda lattice hierarchy I. Proc. Japan Acad. Ser. A 59, 167–170 (1983)Google Scholar
  43. W.
    Walker R.: Algebraic Curves. Springer, New York (1991)Google Scholar
  44. Wi.
    Wilson G.: The modified Lax and two-dimensional Toda lattice equations associated with simple Lie algebras. Ergodic Theory Dynam. Syst. 1, 361–380 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  45. ZMR.
    Zakharov V.E., Musher S.L., Rubenchik A.M.: Weak Langmuir turbulence of an isothermal plasma. Sov. Phys. JETP 42, 80–86 (1975)ADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA
  2. 2.Department of Mathematics and StatisticsUtah State UniversityLoganUSA

Personalised recommendations