Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 9 (2013), 079, 42 pages      arXiv:1108.3587      https://doi.org/10.3842/SIGMA.2013.079
Contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa

A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy

Luc Haine and Didier Vanderstichelen
Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium

Received July 31, 2013, in final form November 30, 2013; Published online December 12, 2013

Abstract
We show that the (semi-infinite) Ablowitz-Ladik (AL) hierarchy admits a centerless Virasoro algebra of master symmetries in the sense of Fuchssteiner [Progr. Theoret. Phys. 70 (1983), 1508-1522]. An explicit expression for these symmetries is given in terms of a slight generalization of the Cantero, Moral and Velázquez (CMV) matrices [Linear Algebra Appl. 362 (2003), 29-56] and their action on the tau-functions of the hierarchy is described. The use of the CMV matrices turns out to be crucial for obtaining a Lax pair representation of the master symmetries. The AL hierarchy seems to be the first example of an integrable hierarchy which admits a full centerless Virasoro algebra of master symmetries, in contrast with the Toda lattice and Korteweg-de Vries hierarchies which possess only ''half of'' a Virasoro algebra of master symmetries, as explained in Adler and van Moerbeke [Duke Math. J. 80 (1995), 863-911], Damianou [Lett. Math. Phys. 20 (1990), 101-112] and Magri and Zubelli [Comm. Math. Phys. 141 (1991), 329-351].

Key words: Ablowitz-Ladik hierarchy; master symmetries; Virasoro algebra.

pdf (588 kb)   tex (39 kb)

References

  1. Ablowitz M.J., Ladik J.F., Nonlinear differential-difference equations, J. Math. Phys. 16 (1975), 598-603.
  2. Ablowitz M.J., Ladik J.F., Nonlinear differential-difference equations and Fourier analysis, J. Math. Phys. 17 (1976), 1011-1018.
  3. Adler M., Shiota T., van Moerbeke P., Random matrices, Virasoro algebras, and noncommutative KP, Duke Math. J. 94 (1998), 379-431, solv-int/9812006.
  4. Adler M., van Moerbeke P., Matrix integrals, Toda symmetries, Virasoro constraints, and orthogonal polynomials, Duke Math. J. 80 (1995), 863-911, solv-int/9706010.
  5. Adler M., van Moerbeke P., Integrals over classical groups, random permutations, Toda and Toeplitz lattices, Comm. Pure Appl. Math. 54 (2001), 153-205, math.CO/9912143.
  6. Adler M., van Moerbeke P., Recursion relations for unitary integrals, combinatorics and the Toeplitz lattice, Comm. Math. Phys. 237 (2003), 397-440, math-ph/0201063.
  7. Aldous D., Diaconis P., Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem, Bull. Amer. Math. Soc. 36 (1999), 413-432.
  8. Bowick M.J., Morozov A., Shevitz D., Reduced unitary matrix models and the hierarchy of τ-functions, Nuclear Phys. B 354 (1991), 496-530.
  9. Cafasso M., Matrix biorthogonal polynomials on the unit circle and non-abelian Ablowitz-Ladik hierarchy, J. Phys. A: Math. Theor. 42 (2009), 365211, 20 pages, arXiv:0804.3572.
  10. Cantero M.J., Moral L., Velázquez L., Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle, Linear Algebra Appl. 362 (2003), 29-56, math.CA/0204300.
  11. Cruz-Barroso R., González-Vera P., A Christoffel-Darboux formula and a Favard's theorem for orthogonal Laurent polynomials on the unit circle, J. Comput. Appl. Math. 179 (2005), 157-173.
  12. Damianou P.A., Master symmetries and R-matrices for the Toda lattice, Lett. Math. Phys. 20 (1990), 101-112.
  13. Date E., Kashiwara M., Jimbo M., Miwa T., Transformation groups for soliton equations, in Nonlinear Integrable Systems - Classical Theory and Quantum Theory (Kyoto, 1981), World Sci. Publishing, Singapore, 1983, 39-119.
  14. Dijkgraaf R., Verlinde H., Verlinde E., Loop equations and Virasoro constraints in nonperturbative two-dimensional quantum gravity, Nuclear Phys. B 348 (1991), 435-456.
  15. Faybusovich L., Gekhtman M., Poisson brackets on rational functions and multi-Hamiltonian structure for integrable lattices, Phys. Lett. A 272 (2000), 236-244, nlin.SI/0006045.
  16. Fernandes R.L., On the master symmetries and bi-Hamiltonian structure of the Toda lattice, J. Phys. A: Math. Gen. 26 (1993), 3797-3803.
  17. Forrester P.J., Witte N.S., Bi-orthogonal polynomials on the unit circle, regular semi-classical weights and integrable systems, Constr. Approx. 24 (2006), 201-237, math.CA/0412394.
  18. Fuchssteiner B., Mastersymmetries, higher order time-dependent symmetries and conserved densities of nonlinear evolution equations, Progr. Theoret. Phys. 70 (1983), 1508-1522.
  19. Fukuma M., Kawai H., Nakayama R., Continuum Schwinger-Dyson equations and universal structures in two-dimensional quantum gravity, Internat. J. Modern Phys. A 6 (1991), 1385-1406.
  20. Gesztesy F., Holden H., Michor J., Teschl G., Local conservation laws and the Hamiltonian formalism for the Ablowitz-Ladik hierarchy, Stud. Appl. Math. 120 (2008), 361-423, arXiv:0711.1644.
  21. Grünbaum F.A., Haine L., A theorem of Bochner, revisited, in Algebraic Aspects of Integrable Systems, Progr. Nonlinear Differential Equations Appl., Vol. 26, Birkhäuser Boston, Boston, MA, 1997, 143-172.
  22. Haine L., Horozov E., Toda orbits of Laguerre polynomials and representations of the Virasoro algebra, Bull. Sci. Math. 117 (1993), 485-518.
  23. Haine L., Semengue J.P., The Jacobi polynomial ensemble and the Painlevé VI equation, J. Math. Phys. 40 (1999), 2117-2134.
  24. Haine L., Vanderstichelen D., A centerless representation of the Virasoro algebra associated with the unitary circular ensemble, J. Comput. Appl. Math. 236 (2011), 19-27, arXiv:1001.4244.
  25. Kac V., Schwarz A., Geometric interpretation of the partition function of 2D gravity, Phys. Lett. B 257 (1991), 329-334.
  26. Kac V.G., Raina A.K., Bombay lectures on highest weight representations of infinite-dimensional Lie algebras, Advanced Series in Mathematical Physics, Vol. 2, World Scientific Publishing Co. Inc., Teaneck, NJ, 1987.
  27. Kharchev S., Mironov A., Integrable structures of unitary matrix models, Internat. J. Modern Phys. A 7 (1992), 4803-4824.
  28. Kharchev S., Mironov A., Zhedanov A., Faces of relativistic Toda chain, Internat. J. Modern Phys. A 12 (1997), 2675-2724, hep-th/9606144.
  29. Kharchev S., Mironov A., Zhedanov A., Different aspects of relativistic Toda chain, in Symmetries and Integrability of Difference Equations (Canterbury, 1996), London Math. Soc. Lecture Note Ser., Vol. 255, Editors P.A. Clarkson, F.W. Nijhoff, Cambridge Univ. Press, Cambridge, 1999, 23-40, hep-th/9612094.
  30. Martinec E.J., On the origin of integrability in matrix models, Comm. Math. Phys. 138 (1991), 437-449.
  31. Mironov A., Morozov A., On the origin of Virasoro constraints in matrix models: Lagrangian approach, Phys. Lett. B 252 (1990), 47-52.
  32. Nenciu I., Lax pairs for the Ablowitz-Ladik system via orthogonal polynomials on the unit circle, Int. Math. Res. Not. 2005 (2005), 647-686, math-ph/0412047.
  33. Rains E.M., Increasing subsequences and the classical groups, Electron. J. Combin. 5 (1998), R12, 9 pages.
  34. Simon B., Orthogonal polynomials on the unit circle. Part 1. Classical theory, American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI, 2005.
  35. Simon B., Orthogonal polynomials on the unit circle. Part 2. Spectral theory, American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI, 2005.
  36. Tracy C.A., Widom H., Fredholm determinants, differential equations and matrix models, Comm. Math. Phys. 163 (1994), 33-72, hep-th/9306042.
  37. Ueno K., Takasaki K., Toda lattice hierarchy, in Group Representations and Systems of Differential Equations (Tokyo, 1982), Adv. Stud. Pure Math., Vol. 4, North-Holland, Amsterdam, 1984, 1-95.
  38. Vanderstichelen D., Virasoro symmetries for the Ablowitz-Ladik hierarchy and non-intersecting Brownian motion models, Ph.D. Thesis, Université Catholique de Louvain, 2011.
  39. Zubelli J.P., Magri F., Differential equations in the spectral parameter, Darboux transformations and a hierarchy of master symmetries for KdV, Comm. Math. Phys. 141 (1991), 329-351.

Previous article  Next article   Contents of Volume 9 (2013)