Ils donnent deux structures de Poisson pour lesquelles le réseau de Schur est hamiltonien :
– {αi, αi+1}ℓ= −(1−α2i)(1−α2i+1), pour laquelle la fonction Hℓ:= n−1
X
i=0
ln(1+αi)
est un hamiltonien du système, – {αi, αj}nℓ= −(1−αi)(1−αj) j−1 Y k=i+1 1 − αk 1 + αk
, où i < j, pour laquelle la fonction Hnℓ:=
n
X
i=0
αiαi−1 est un hamiltonien du système.
Ils donnent également l’équation de Lax suivante pour ce système : ˙
U = [P+(U + U−1), U ], (4.37)
où U est la matrice
U := −α0α1 −α0β12α2 −α0β12. . . β2n−1αn 1 −α1α2 1 1 −αn−1αn (0) , (4.38)
où pour tout k = 1 . . . n − 1, on a β2
k = 1 − α2k.
On va à présent utiliser les résultats obtenus sur le réseau de Schur modifié pour retrouver cette équation de Lax. On rappelle que le réseau de Schur modifié est défini par
˙αk= (1 − |αk|2)(αk+1− αk−1), (4.39)
où pour tout k = 1 . . . n − 1, αk∈ C, α0= 1 et |αn| = 1. De plus l’équation de Lax
˙
H = [H, S(H)], (4.40) correspond au réseau de Schur modifié, avec H ∈ Hα
n de la forme H := −α0α1 −α0β1α2 −α0β1β2α3 −α0β1. . . βn−1αn β1 −α1α2 −α1β2α3 −α1β2. . . βn−1αn −αn−2βn−1αn βn−1 −αn−1αn (0) , (4.41) et avec S(H) = 1 2(−α0α1− (−α0α1)) −β1 β1 −βn−1 βn−1 12(−αn−1αn− (−αn−1αn)) (0) (0) .
On remarque alors que l’on peut se restreindre à l’ensemble {αk ∈ R, k = 0 . . . n}
et on a alors U = η−1H
Rη, où η est la matrice diagonale η := diag(η1, . . . , ηn) avec ηk+1
ηk = βk. De plus, la restriction de S(H) est
S(H)R:= 0 −β1 β1 −βn−1 βn−1 0 (0) (0) , ou encore S(H)R= P−(HR) − P+(HR−1).
On constate également que le réseau de Schur modifié devient le réseau de Schur (réel). Par contre comme on ne se restreint pas à une sous-variété de Poisson, il n’y a plus de structure de Poisson sur les matrices HR, mais de toute façon la restriction
de l’hamiltonien T du réseau de Schur modifié s’annule.
Cependant l’équation de Lax ˙H = [H, S(H)] du réseau de Schur modifié donne par restriction l’équation de Lax
˙
HR= [HR, S(H)R]
correspondant au réseau de Schur. On vérifie alors que cette équation de Lax est la même que celle de l’articleOn Schur flows [FG99] à conjugaison près.
En effet, comme U = η−1H Rη, on a ˙ U = −η−1˙ηη−1HRη + η−1H˙Rη + η−1HR˙η = −η−1˙ηU + η−1[HR, S(H)R]η + U η−1˙η = [U, η−1˙η] + [U, η−1(P−(HR) − P+(HR−1))η] = [U, η−1˙η + P−(U ) − P+(U−1)],
or, de la même façon que dans le lemme 3.3.9, en utilisant le fait que (η−1˙η)
k+1,k+1−
(η−1˙η)
k,k= Uk+1,k+1− Uk,k, on en déduit que [U, η−1˙η] = [U, P0(U )]. Ainsi, on a
˙
U = [U, P0(U ) + P−(U ) − P+(U−1)]
= [U, U − P+(U ) − P+(U−1)]
= [P+(U + U−1), U ],
et on retrouve l’équation de Lax du réseau de Schur donnée par Faybusovich et Gekhtman [FG99].
Bibliographie
[Adl79] M. Adler. On a trace functional for formal pseudodifferential operators and the Hamiltonian structure of Korteweg-de Vries type equations. In Global analysis (Proc. Biennial Sem. Canad. Math. Congr., Univ. Cal- gary, Calgary, Alta., 1978), volume 755 of Lecture Notes in Math., pages 1–16. Springer, Berlin, 1979.
[AG94] Gregory S. Ammar and William B. Gragg. Schur flows for orthogonal Hessenberg matrices. InHamiltonian and gradient flows, algorithms and control, volume 3 of Fields Inst. Commun., pages 27–34. Amer. Math. Soc., Providence, RI, 1994.
[AL75] M. J. Ablowitz and J. F. Ladik. Nonlinear differential-difference equa- tions. J. Mathematical Phys., 16 :598–603, 1975.
[AM78] Ralph Abraham and Jerrold E. Marsden. Foundations of mechanics. Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, Reading, Mass., 1978. Second edition, revised and enlarged, With the assistance of Tudor Raţiu and Richard Cushman.
[Arn78] V. I. Arnold. Mathematical methods of classical mechanics. Springer- Verlag, New York, 1978. Translated from the Russian by K. Vogtmann and A. Weinstein, Graduate Texts in Mathematics, 60.
[AvM01] Mark Adler and Pierre van Moerbeke. Integrals over classical groups, random permutations, Toda and Toeplitz lattices. Comm. Pure Appl. Math., 54(2) :153–205, 2001.
[AvM03] M. Adler and P. van Moerbeke. Recursion relations for unitary in- tegrals, combinatorics and the Toeplitz lattice. Comm. Math. Phys., 237(3) :397–440, 2003.
[AvMV04] Mark Adler, Pierre van Moerbeke, and Pol Vanhaecke. Algebraic inte- grability, Painlevé geometry and Lie algebras, volume 47 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 2004.
[AvMV08] Mark Adler, Pierre van Moerbeke, and Pol Vanhaecke. Singularity confinement for a class of m-th order difference equations of combi- natorics. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366(1867) :877–922, 2008.
[BA09] Khaoula Ben Abdeljelil. L’intégrabilité du réseau de 2-Toda sur sln(C)
et sa généralisation aux algèbres de Lie semi-simples.C. R. Math. Acad. Sci. Paris, 347(15-16) :943–946, 2009.
[BA11] Khaoula Ben Abdeljelil. The integrability of the 2-Toda lattice on a simple Lie algebra. J. Geom. Phys., 61(10) :2015–2032, 2011.
[Bum04] Daniel Bump. Lie groups, volume 225 of Graduate Texts in Mathema- tics. Springer-Verlag, New York, 2004.
[BV88] K. H. Bhaskara and K. Viswanath. Poisson algebras and Poisson ma- nifolds, volume 174 of Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Harlow, 1988.
[CdSW99] Ana Cannas da Silva and Alan Weinstein. Geometric models for noncommutative algebras, volume 10 of Berkeley Mathematics Lecture Notes. American Mathematical Society, Providence, RI, 1999.
[CH90] A. K. Common and S. T. Hafez. Continued-fraction solutions to the Riccati equation and integrable lattice systems.J. Phys. A, 23(4) :455– 466, 1990.
[Chu84] Moody T. Chu. On the global convergence of the Toda lattice for real normal matrices and its applications to the eigenvalue problem. SIAM J. Math. Anal., 15(1) :98–104, 1984.
[Com92] A. K. Common. A solution of the initial value problem for half-infinite integrable lattice systems. Inverse Problems, 8(3) :393–408, 1992. [DK00] J. J. Duistermaat and J. A. C. Kolk.Lie groups. Universitext. Springer-
Verlag, Berlin, 2000.
[DL91] Percy Deift and Luen Chau Li. Poisson geometry of the analog of the Miura maps and Bäcklund-Darboux transformations for equations of Toda type and periodic Toda flows. Comm. Math. Phys., 143(1) :201– 214, 1991.
[DLF02] Pantelis A. Damianou and Rui Loja Fernandes. From the Toda lattice to the Volterra lattice and back. Rep. Math. Phys., 50(3) :361–378, 2002.
[DLNT86] P. Deift, L. C. Li, T. Nanda, and C. Tomei. The Toda flow on a generic orbit is integrable. Comm. Pure Appl. Math., 39(2) :183–232, 1986. [DLT89] P. Deift, L. C. Li, and C. Tomei. Matrix factorizations and integrable
systems. Comm. Pure Appl. Math., 42(4) :443–521, 1989.
[Dri83] V. G. Drinfel′d. Hamiltonian structures on Lie groups, Lie bialgebras
and the geometric meaning of classical Yang-Baxter equations. Dokl. Akad. Nauk SSSR, 268(2) :285–287, 1983.
[DZ05] Jean-Paul Dufour and Nguyen Tien Zung. Poisson structures and their normal forms, volume 242 of Progress in Mathematics. Birkhäuser Ver- lag, Basel, 2005.
[EG95] Amine M. El Gradechi. Geometric quantization of an OSp(1/2) coad- joint orbit. Lett. Math. Phys., 35(1) :13–26, 1995.
[FG99] Leonid Faybusovich and Michael Gekhtman. On Schur flows. J. Phys. A, 32(25) :4671–4680, 1999.
[FH91] William Fulton and Joe Harris. Representation theory, volume 129 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. A first course, Readings in Mathematics.
[FV01] Rui Loja Fernandes and Pol Vanhaecke. Hyperelliptic Prym varieties and integrable systems. Comm. Math. Phys., 221(1) :169–196, 2001. [Gav99] Lubomir Gavrilov. Generalized Jacobians of spectral curves and com-
pletely integrable systems. Math. Z., 230(3) :487–508, 1999.
[Gek93] M. I. Gekhtman. Nonabelian nonlinear lattice equations on finite inter- val. J. Phys. A, 26(22) :6303–6317, 1993.
Bibliographie 125 [GZ98] Lubomir Gavrilov and Angel Zhivkov. The complex geometry of the
Lagrange top. Enseign. Math. (2), 44(1-2) :133–170, 1998.
[Hum78] James E. Humphreys. Introduction to Lie algebras and representation theory, volume 9 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1978. Second printing, revised.
[KS97] Y. Kosmann-Schwarzbach. Lie bialgebras, Poisson Lie groups and dres- sing transformations. InIntegrability of nonlinear systems (Pondicherry, 1996), volume 495 of Lecture Notes in Phys., pages 104–170. Springer, Berlin, 1997.
[LGMV11] Camille Laurent-Gengoux, Eva Miranda, and Pol Vanhaecke. Action- angle coordinates for integrable systems on Poisson manifolds. Int. Math. Res. Not. IMRN, (8) :1839–1869, 2011.
[LGPV12] Camille Laurent-Gengoux, Anne Pichereau, and Pol Vanhaecke.Poisson structures, volume 347 of Die Grundlehren der mathematischen Wissen- schaften. Springer-Verlag, Berlin, 2012.
[LGSX08] Camille Laurent-Gengoux, Mathieu Stiénon, and Ping Xu. Holomorphic Poisson manifolds and holomorphic Lie algebroids.Int. Math. Res. Not. IMRN, pages Art. ID rnn 088, 46, 2008.
[Lic77] André Lichnerowicz. Les variétés de Poisson et leurs algèbres de Lie associées. J. Differential Geometry, 12(2) :253–300, 1977.
[LP89] Luen Chau Li and Serge Parmentier. Nonlinear Poisson structures and r-matrices. Comm. Math. Phys., 125(4) :545–563, 1989.
[Lu90] Jiang-Hua Lu. Multiplicative and affine Poisson structures on Lie groups. ProQuest LLC, Ann Arbor, MI, 1990. Thesis (Ph.D.)–University of California, Berkeley.
[Lu91] Jiang-Hua Lu. Momentum mappings and reduction of Poisson actions. In Symplectic geometry, groupoids, and integrable systems (Berkeley, CA, 1989), volume 20 of Math. Sci. Res. Inst. Publ., pages 209–226. Springer, New York, 1991.
[Lu99] Jiang-Hua Lu. Coordinates on Schubert cells, Kostant’s harmonic forms, and the Bruhat Poisson structure on G/B. Transform. Groups, 4(4) :355–374, 1999.
[LW90] Jiang-Hua Lu and Alan Weinstein. Poisson Lie groups, dressing transformations, and Bruhat decompositions. J. Differential Geom., 31(2) :501–526, 1990.
[Mit95] D. A. Mit’kin. On an estimate for the number of solutions of some “jagged” systems of equations. Mat. Zametki, 57(5) :681–687, 797, 1995. [OR04] Juan-Pablo Ortega and Tudor S. Ratiu. Momentum maps and Hamil- tonian reduction, volume 222 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 2004.
[Ped02] M. Pedroni. Bi-Hamiltonian aspects of the separability of variables in the Neumann system. Teoret. Mat. Fiz., 133(3) :475–484, 2002. [PV98] M. Pedroni and P. Vanhaecke. A Lie algebraic generalization of the
Mumford system, its symmetries and its multi-Hamiltonian structure. Regul. Chaotic Dyn., 3(3) :132–160, 1998. J. Moser at 70.
[STS83] M. Semenov, Tian, and Shansky. What a classical r-matrix is. Functio- nal Anal. Appl., 17(4) :259–272, 1983.
[Vai94] Izu Vaisman. Lectures on the geometry of Poisson manifolds, volume 118 ofProgress in Mathematics. Birkhäuser Verlag, Basel, 1994.
[Van01] Pol Vanhaecke. Integrable systems in the realm of algebraic geometry, volume 1638 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, second edition, 2001.
[vM06] Pierre van Moerbeke. Combinatorics and integrable geometry. InBili- near integrable systems : from classical to quantum, continuous to dis- crete, volume 201 of NATO Sci. Ser. II Math. Phys. Chem., pages 335– 362. Springer, Dordrecht, 2006.
[War83] Frank W. Warner. Foundations of differentiable manifolds and Lie groups, volume 94 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1983. Corrected reprint of the 1971 edition.
[Wei83] Alan Weinstein. The local structure of Poisson manifolds.J. Differential Geom., 18(3) :523–557, 1983.
[Wel80] R. O. Wells, Jr. Differential analysis on complex manifolds, volume 65 ofGraduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1980.
[Whi88] E. T. Whittaker. A treatise on the analytical dynamics of particles and rigid bodies. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1988. With an introduction to the problem of three bodies, Reprint of the 1937 edition, With a foreword by William Mc- Crea.
[Xu03] Ping Xu. Dirac submanifolds and Poisson involutions. Ann. Sci. École Norm. Sup. (4), 36(3) :403–430, 2003.