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A Class of Integrable Flows on the Space of Symmetric Matrices

Bloch, Anthony M. and Brînzănescu, Vasile and Iserles, Arieh and Marsden, Jerrold E. and Ratiu, Tudor S. (2009) A Class of Integrable Flows on the Space of Symmetric Matrices. Communications in Mathematical Physics, 290 (2). pp. 399-435. ISSN 0010-3616.

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For a given skew symmetric real n × n matrix N, the bracket [X, Y]_N = XNY − YNX defines a Lie algebra structure on the space Sym(n, N) of symmetric n × n real matrices and hence a corresponding Lie-Poisson structure. The purpose of this paper is to investigate the geometry, integrability, and linearizability of the Hamiltonian system X[overdot] = [X^2, N], or equivalently in Lax form, the equation X[overdot] = [X, XN + NX] on this space along with a detailed study of the Poisson geometry itself. If N has distinct eigenvalues, it is proved that this system is integrable on a generic symplectic leaf of the Lie-Poisson structure of Sym(n, N). This is established by finding another compatible Poisson structure. If N is invertible, several remarkable identifications can be implemented. First, (Sym(n, N), [·, ·]) is Lie algebra isomorphic with the symplectic Lie algebra sp (n, N^-1)associated to the symplectic form on R^n given by N^−1. In this case, the system is the reduction of the geodesic flow of the left invariant Frobenius metric on the underlying symplectic group Sp(n, N^−1). Second, the trace of the product of matrices defines a non-invariant non-degenerate inner product on Sym(n, N) which identifies it with its dual. Therefore Sym(n, N) carries a natural Lie-Poisson structure as well as a compatible “frozen bracket” structure. The Poisson diffeomorphism from Sym(n, N) to sp(n, N^-1) maps our system to a Mischenko-Fomenko system, thereby providing another proof of its integrability if N is invertible with distinct eigenvalues. Third, there is a second ad-invariant inner product on Sym(n, N); using it to identify Sym(n, N) with itself and composing it with the dual of the Lie algebra isomorphism with sp(n, N^-1), our system becomes a Mischenko- Fomenko system directly on Sym(n, N). If N is invertible and has distinct eigenvalues, it is shown that this geodesic flow on Sym(n, N) is linearized on the Prym subvariety of the Jacobian of the spectral curve associated to a Lax pair formulation with parameter of the system. If, on the other hand, N has nullity one and distinct eigenvalues, in spite of the fact that the system is completely integrable, it is shown that the flow does not linearize on the Jacobian of the spectral curve.

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Additional Information:© 2009 Springer. Received: 21 February 2007; accepted: 30 March 2009; published online: 8 July 2009. Research partially supported by NSF grants CMS-0408542 and DMS-0604307. Research partially supported by the Swiss SCOPES grant IB7320-110721/1, 2005-2008, and MEdC Contract 2-CEx 06-11-22/25.07.2006. Research partially supported by the California Institute of Technology and NSF-ITR Grant ACI-0204932. Research partially supported by the Swiss NSF and the Swiss SCOPES grant IB7320-110721/1. We thank G. Prasad for his observation regarding Lie algebras. Luc Haine and Pol Vanhaecke have our gratitude for many very illuminating discussions regarding algebraic complete integrability. We also thank Alexey Bolsinov, Percy Deift, Igor Dolgachev, Michael Gekhtman, Rob Lazarsfeld, Alejandro Uribe, and Nguyen Tien Zung for useful conversations that clarified various points in the paper and thereby improved our exposition. Finally we would like to thank the referee for an extraordinarily useful and insightful referee report and for pointing out reference and we would also like to thank the editor for his very helpful input.
Funding AgencyGrant Number
Swiss SCOPESIB7320-110721/1
MEdC2-CEx 06-11-22/25.07.2006
Swiss SCOPESIB7320-110721/1
Swiss National Science FoundationUNSPECIFIED
Record Number:CaltechAUTHORS:20090810-113731412
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:14925
Deposited By: Jason Perez
Deposited On:10 Aug 2009 22:50
Last Modified:26 Dec 2012 11:10

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