Inverse moment problem for non-Abelian Coxeter double Bruhat cells

Abstract

We solve the inverse problem for non-Abelian Coxeter double Bruhat cells in terms of the matrix Weyl functions. This result can be used to establish complete integrability of the non-Abelian version of nonlinear Coxeter-Toda lattices in $GL_n$.

Key words: Non-Abelian lattices, Coxeter double Bruhat cells, inverse problems.

Full Text

Article Information

Title	Inverse moment problem for non-Abelian Coxeter double Bruhat cells
Source	Methods Funct. Anal. Topology, Vol. 22 (2016), no. 2, 117-136
MathSciNet	MR3522855
zbMATH	06665383
Milestones	Received 04/05/2015; Revised 09/02/2016
Copyright	The Author(s) 2016 (CC BY-SA)

Authors Information

Michael Gekhtman
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA

Citation Example

Michael Gekhtman, Inverse moment problem for non-Abelian Coxeter double Bruhat cells, Methods Funct. Anal. Topology 22 (2016), no. 2, 117-136.

BibTex

@article {MFAT846,
    AUTHOR = {Gekhtman, Michael},
     TITLE = {Inverse moment problem for non-Abelian Coxeter double Bruhat cells},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {22},
      YEAR = {2016},
    NUMBER = {2},
     PAGES = {117-136},
      ISSN = {1029-3531},
  MRNUMBER = {MR3522855},
 ZBLNUMBER = {06665383},
       URL = {http://mfat.imath.kiev.ua/article/?id=846},
}

References

1. N. I. Akhiezer, The classical moment problem and some related questions in analysis, Translated by N. Kemmer, Hafner Publishing Co., New York, 1965.  MathSciNet
2. Michael Atiyah and Nigel Hitchin, The geometry and dynamics of magnetic monopoles, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1988.  MathSciNet CrossRef
3. Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), no. 1, 1-52.  MathSciNet CrossRef
4. Arkady Berenstein and Vladimir Retakh, Noncommutative double Bruhat cells and their factorizations, Int. Math. Res. Not. (2005), no. 8, 477-516.  MathSciNet CrossRef
5. Ju. M. Berezans′kii, Expansions in eigenfunctions of selfadjoint operators, American Mathematical Society, Providence, R.I., 1968.  MathSciNet
6. Yu. M. Berezanskii, M. I. Gekhtman, and M. E. Shmoish, Integration of certain chains of nonlinear difference equations by the method of the inverse spectral problem, Ukrainian Math. J. 38 (1986), no. 1, 74-78.  MathSciNet CrossRef
7. Yu. M. Berezanskii and A. A. Mokhon′ko, Integration of some nonlinear differential-difference equations using the spectral theory of normal block-Jacobi matrices, Funct. Anal. Appl. 42 (2008), no. 1, 1-18.  MathSciNet CrossRef
8. Yu. M. Berezansky, I. Ya. Ivasiuk, and O. A. Mokhonko, Recursion relation for orthogonal polynomials on the complex plane, Methods Funct. Anal. Topology 14 (2008), no. 2, 108-116.  MathSciNet MFAT Article
9. M. Bruschi, S. V. Manakov, O. Ragnisco, D. Levi, The nonabelian Toda lattice (discrete analogue of the matrix Schrodinger spectral problem), J. Math. Phys. 21 (1980), 2749-2753.
10. M. J. Cantero, L. Moral, and L. Velazquez, Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle, Linear Algebra Appl. 362 (2003), 29-56.  MathSciNet CrossRef
11. Philippe Di Francesco and Rinat Kedem, $Q$-systems, heaps, paths and cluster positivity, Comm. Math. Phys. 293 (2010), no. 3, 727-802.  MathSciNet CrossRef
12. Philippe Di Francesco and Rinat Kedem, Non-commutative integrability, paths and quasi-determinants, Adv. Math. 228 (2011), no. 1, 97-152.  MathSciNet CrossRef
13. Pavel Etingof, Israel Gelfand, and Vladimir Retakh, Factorization of differential operators, quasideterminants, and nonabelian Toda field equations, Math. Res. Lett. 4 (1997), no. 2-3, 413-425.  MathSciNet CrossRef
14. Ludwig D. Faddeev and Leon A. Takhtajan, Hamiltonian methods in the theory of solitons, Classics in Mathematics, Springer, Berlin, 2007.  MathSciNet
15. Leonid Faybusovich and Michael Gekhtman, Elementary Toda orbits and integrable lattices, J. Math. Phys. 41 (2000), no. 5, 2905-2921.  MathSciNet CrossRef
16. Leonid Faybusovich and Michael Gekhtman, Poisson brackets on rational functions and multi-Hamiltonian structure for integrable lattices, Phys. Lett. A 272 (2000), no. 4, 236-244.  MathSciNet CrossRef
17. Leonid Faybusovich and Michael Gekhtman, Inverse moment problem for elementary co-adjoint orbits, Inverse Problems 17 (2001), no. 5, 1295-1306.  MathSciNet CrossRef
18. Shaun M. Fallat, Bidiagonal factorizations of totally nonnegative matrices, Amer. Math. Monthly 108 (2001), no. 8, 697-712.  MathSciNet CrossRef
19. Sergey Fomin and Andrei Zelevinsky, Double Bruhat cells and total positivity, J. Amer. Math. Soc. 12 (1999), no. 2, 335-380.  MathSciNet CrossRef
20. Sergey Fomin and Andrei Zelevinsky, Total positivity: tests and parametrizations, Math. Intelligencer 22 (2000), no. 1, 23-33.  MathSciNet CrossRef
21. Israel Gelfand, Sergei Gelfand, Vladimir Retakh, and Robert Lee Wilson, Quasideterminants, Adv. Math. 193 (2005), no. 1, 56-141.  MathSciNet CrossRef
22. M. I. Gekhtman, Integration of non-Abelian Toda-type chains, Funct. Anal. Appl. 24 (1990), no. 3, 231-233.  MathSciNet CrossRef
23. Michael Gekhtman and Olena Korovnichenko, Matrix Weyl functions and non-abelian Coxeter-Toda lattices, Notions of positivity and the geometry of polynomials, Trends Math., Birkhauser/Springer Basel AG, Basel, 2011, pp. 221-237.  MathSciNet CrossRef
24. M. Gekhtman, Hamiltonian structure of non-abelian Toda lattice, Lett. Math. Phys. 46 (1998), no. 3, 189-205.  MathSciNet CrossRef
25. Michael Gekhtman, Michael Shapiro, and Alek Vainshtein, Cluster algebras and Poisson geometry, Mathematical Surveys and Monographs, vol. 167, American Mathematical Society, Providence, RI, 2010.  MathSciNet CrossRef
26. Michael Gekhtman, Michael Shapiro, and Alek Vainshtein, Generalized Backlund-Darboux transformations for Coxeter-Toda flows from a cluster algebra perspective, Acta Math. 206 (2011), no. 2, 245-310.  MathSciNet CrossRef
27. Tim Hoffmann, Johannes Kellendonk, Nadja Kutz, and Nicolai Reshetikhin, Factorization dynamics and Coxeter-Toda lattices, Comm. Math. Phys. 212 (2000), no. 2, 297-321.  MathSciNet CrossRef
28. S. Kharchev, A. Mironov, and A. Zhedanov, Faces of relativistic Toda chain, Internat. J. Modern Phys. A 12 (1997), no. 15, 2675-2724.  MathSciNet CrossRef
29. Bertram Kostant, The solution to a generalized Toda lattice and representation theory, Adv. in Math. 34 (1979), no. 3, 195-338.  MathSciNet CrossRef
30. M. Krein, Infinite $J$-matrices and a matrix-moment problem, Doklady Akad. Nauk SSSR (N.S.) 69 (1949), 125-128.  MathSciNet
31. I. Krichever, The periodic nonabelian Toda chain and its two-dimensional generalization, Russ. Math. Surveys 39 (1981), 32-81.
32. Jurgen Moser, 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 Phys., Vol. 38, Springer, Berlin, 1975, pp. 467-497.  MathSciNet
33. Michael Shmoish, On generalized spectral functions, the parametrization of block Hankel and block Jacobi matrices, and some root location problems, Linear Algebra Appl. 202 (1994), 91-128.  MathSciNet CrossRef
34. W. W. Symes, Systems of Toda type, inverse spectral problems, and representation theory, Invent. Math. 59 (1980), no. 1, 13-51.  MathSciNet CrossRef