Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0809.4650

Completeness of determinantal Hamiltonian flows on the matrix affine Poisson space

Michael Gekhtman·Milen Yakimov
View original source ↗Read source document ↗Export metadata ↓

Abstract

The matrix affine Poisson space (M_{m,n}, pi_{m,n}) is the space of complex rectangular matrices equipped with a canonical quadratic Poisson structure which in the square case m=n reduces to the standard Poisson structure on GL_n(C). We prove that the Hamiltonian flows of all minors are complete. As a corollary we obtain that all Kogan-Zelevinsky integrable systems on M_{n,n} are complete and thus induce (analytic) Hamiltonian actions of C^{n(n-1)/2} on (M_{n,n}, pi_{n,n}) (as well as on GL_n(C) and on SL_n(C)). We define Gelfand-Zeitlin integrable systems on (M_{n,n}, pi_{n,n}) from chains of Poisson projections and prove that their flows are also complete. This is an analog for the quadratic Poisson structure pi_{n,n} of the recent result of Kostant and Wallach [KW] that the flows of the complexified classical Gelfand-Zeitlin integrable systems are complete.

Explore related subjects

math.SGmath.QA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michael Gekhtman, Milen Yakimov. 2008-09-26. Completeness of determinantal Hamiltonian flows on the matrix affine Poisson space. https://doi.org/10.1007/s11005-009-0337-0

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Darboux's Theorem in $p$-adic symplectic geometry

We prove a non Archimedean Darboux's Theorem: any two symplectic forms on a $p$-adic analytic manifold are locally isomorphic. Understanding local problems such as the existence of flows or the normalization of singularities in the theory of integrable systems, is essential to understand the physics behind these systems. Our result tells us that the phase space defined by a $p$-adic manifold is locally standard, allowing us to concentrate on the equations defining the dynamics rather than on the space itself. Our proof uses a non Archimedean version of Moser's Path Method to push one symplectic form onto another one by a flow. A central technical contribution of the paper is the proof that the flow is given by a power series with non zero radius of convergence, which requires geometric analytic estimates and does not follow from algebraic considerations. The main difficulty of the paper is analytic: an ODE will not have a long-time solution even in a neighborhood of a fixed point but we are able to find a condition for a long-time solution to exist, which allows us to tweak the ODE in Moser's Path Method so that it satifies the condition. As a global application, we derive a classification of second-countable $p$-adic analytic symplectic manifolds in terms of $p$-adic volume, which generalizes a classical theorem of J-P. Serre.

math.SG↗
arXiv2026

A Multidimensional Birkhoff Theorem for some $C^0$ Lagrangians

We prove a multidimensional Birkhoff theorem for a new class of "$C^0$ Lagrangian subsets" in cotangent bundles, obtained as Hausdorff limits of compact exact Lagrangian submanifolds with control on their Liouville primitives. If the successive images of such a subset $L$ under the flow of a Tonelli Hamiltonian admit convergent subsequences in both positive and negative time, then $L$ and all its images are Lipschitz graphs over the base. This extends Birkhoff's celebrated theorem for twist maps of the annulus and its known higher-dimensional generalizations, and provides a first version of such results for merely continuous objects. The proof combines Floer-theoretic graph selectors, variational solutions of the Hamilton-Jacobi equation, and weak KAM theory. We also investigate the rigidity and uniqueness of limiting primitives for this new class of singular Lagrangian subsets, which may be of independent interest in $C^0$ symplectic topology.

math.SG↗
arXiv2026

The Legendrian Hopf Link has exactly two Lagrangian fillings

We prove that there are precisely two embedded exact Lagrangian fillings of the standard Legendrian Hopf link, up to compactly supported Hamiltonian isotopy. It was known that the standard Legendrian Hopf link admitted at least two such Lagrangian fillings: we show these are all. Specifically, we use a type of neck-stretching procedure to construct a pseudoholomorphic conic fibration that makes a given arbitrary exact Lagrangian filling fiber over a real curve, under a global pseudoholomorphic Lefschetz fibration. This then allows for an explicit Hamiltonian isotopy to be constructed from any given Lagrangian filling to one of two known standard fillings.

math.SG↗