Inventiones mathematicae

, Volume 113, Issue 1, pp 511–529 | Cite as

A Schur-Horn-Kostant convexity theorem for the diffeomorphism group of the annulus

  • A. M. Bloch
  • H. Flaschka
  • T. Ratiu


The group of area preserving diffeomorphisms of the annulus acts on its Lie algebra, the globally Hamiltonian vectorfields on the annulus. We consider a certain Hilbert space completion of this group (thinking of it as a group of unitary operators induced by the diffeomorphisms), and prove that the projection of an adjoint orbit onto a “Cartan” subalgebra isomorphic toL2 ([0, 1]) is an infinite-dimensional, weakly compact, convex set, whose extreme points coincide with the orbit, through a certain function, of the “permutation” semigroup of measure preserving transformations of [0, 1].


Hilbert Space Extreme Point Unitary Operator Measure Preserve Area Preserve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Akh] Akhiezer, N.: The Classical Moment Problem. (Univ. Math. Monogr.) Edinburgh London: Oliver & Boyd 1965Google Scholar
  2. [A1] Alpern, S.: Approximation to and by measure preserving homeomorphisms. J. Lond. Math. Soc.18, 305–315 (1978)Google Scholar
  3. [AP] Atiyah, M.F., Pressley, A.N.: Convexity and loop groups. In: Artin, M., Tate, J. (eds.) Arithmetic and Geometry: papers dedicated to I.R. Shafarevich on the occasion of his sixtieth birthday, vol. 2. Boston Basel Stuttgart: Birkhaüser 1983Google Scholar
  4. [BR] Bao, D., Ratiu, T.: A candidate maximal torus in infinite dimensions. Contemp. Math., vol.132, pp. 117–124. Providence, RI: Am. Math. Soc. 1992, details to appear in another paperGoogle Scholar
  5. [Bi] Birkhoff, G.: Three observations on linear algebra. Rev. Univ. Nac. Tucumán, Ser. A5, 147–151 (1946) (we have only seen the Math. Reviews summary of this paper)Google Scholar
  6. [BBKR] Bloch, A.M., Brockett, R. W., Kodama, Y., Ratiu, T.: Spectral equations for the long wave limit of the Toda lattice equations. In: Proceedings of the CRM Workshop on Hamiltonian Systems, Transformation Groups and Spectral Transform Methods, pp. 97–402. Harnad, J., Marsden, J.E. (eds.), Publ. CRM, Montréal 1990Google Scholar
  7. [BBR] Bloch, A.M., Brockett, R. W., Ratiu, T.: A new formulation of the generalized Toda lattice equations and their fixed point analysis via the momentum map. Bull.23, 447–456 (1990)Google Scholar
  8. [Bo] Bogojavlenskiî, O.I.: Lax representations with spectral parameter for some dynamical systems. Izv. Akad. Nauk. SSSR52, 243–266 (1988)Google Scholar
  9. [BB] Brockett, R. W., Block, A.: Sorting with the dispersionless limit of the Toda lattice. In: Proceedings of the CRM Workshop on Hamiltonian Systems, Transformation Groups and Spectral Transform Methods, pp. 103–112, Harnad, J., Marsden, J.E. (eds.) Publ. CRM, Montréal 1990Google Scholar
  10. [Bro] Brockett, R. W.: Dynamical systems that sort lists, solve linear programming problems and diagonalize symmetric matrices. Proc. 1988 IEEE Conference on Decision and Control, 1988. Linear Algebra Appl.122–124, pp. 761–777 (1989)Google Scholar
  11. [Br] Brown, J.R.: Approximation theorems for Markov operators. Pac. J. Math.16, 13–23 (1966)Google Scholar
  12. [EM] Ebin, D.G., Marsden, J.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math.92, 102–163 (1970)Google Scholar
  13. [Ha1] Halmos, P.: Approximation theorems for measure preserving transformations. Trans. Am. Math. Soc.55, 1–18 (1944)Google Scholar
  14. [Ha2] Halmos, P.: Measure Theory. Princeton, NJ: D. Van Nostrand 1950Google Scholar
  15. [HLP] Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge: Cambridge University Press 1934Google Scholar
  16. [H] Horn, A.: Doubly stochastic matrices and the diagonal of a rotation matrix. Am. J. Math.76 620–630 (1954)Google Scholar
  17. [K] Kostant, B.: On convexity, the Weyl group and the Iwasawa decomposition. Ann. Sci. Éc. Norm. Supér.6, 413–455 (1973)Google Scholar
  18. [KP] Kac, V.G., Peterson, D.H.: Unitary structure in representations of infinite-dimensional groups and a convexity theorem. Invent. Math.76, 1–14 (1984)Google Scholar
  19. [MO] Marshall, A. W., Olkin, I.: Inequalities: Theory of Majorization and its Applications. New York: Academic Press 1979Google Scholar
  20. [Mo] Moser, J.: Remark on the smooth approximation of volume-preserving homeomorphisms by symplectic diffeomorphisms. Zürich: Forschungsinstitut für Mathematik ETH (Preprint, October 1992)Google Scholar
  21. [Ro] Royden, H.: Real Analysis. New York: Macmillan 1963Google Scholar
  22. [R1] Ryff, J. V.: On the representation of doubly stochastic operators. Pac. J. Math.13, 1379–1386 (1963)Google Scholar
  23. [R2] Ryff, J. V.: Orbits ofL 1 functions under doubly stochastic transformations. Trans. Am. Math. Soc.117, 92–100 (1965)Google Scholar
  24. [R3] Ryff, J. V.: Extreme points of some convex subsets ofL 1(0, 1). Proc. Am. Math. Soc.18, 1026–1034 (1967)Google Scholar
  25. [S] Saks, S.: Theory of the Integral, 2nd ed. (Dover Publ.) New York: Dover 1964Google Scholar
  26. [Sa1] Savieliev, M.V.: Integro-differential nonlinear equations and continual Lie algebras. Commun. Math. Phys.121, 283–290 (1989)Google Scholar
  27. [Sa2] Saveliev, M. V.: On the integrability problem of the continuous long wave approximation of the Toda lattice. Lyon: Éc. Norm. Supé. Preprint (1992)Google Scholar
  28. [SV] Saveliev, M. V., Vershik, A.M.: New examples of continuum graded Lie algebras. Phys. Lett.A143, 121–128 (1990)Google Scholar
  29. [Sch] Schur, I.: Über eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie. Sitzungsber. Berl. Math. Ges.22, 9–20 (1923)Google Scholar
  30. [T] Toda, M.: Theory of Nonlinear Lattices. Berlin Heidelberg New York: Springer 1981Google Scholar
  31. [VGG] Vershik, A.M., Gel'fand, I.M., Graev, M.I.: Representation of the group of diffeomorphisms. Usp. Mat. Nauk30 (no. 6), 1–50 (1975)Google Scholar
  32. [WE] Watson, D.S., Elsner, L.: On Rutishauser's approach to selfsimilar flows. SIAM J. Matrix. Anal. Appl.11, 301–311 (1990)Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • A. M. Bloch
    • 1
  • H. Flaschka
    • 2
  • T. Ratiu
    • 3
  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA
  2. 2.Department of MathematicsThe University of ArizonaTucsonUSA
  3. 3.Department of MathematicsUniversity of CaliforniaSanta CruzUSA

Personalised recommendations