Non-linear Grassmannians as coadjoint orbits

Haller, Stefan; Vizman, Cornelia

doi:10.1007/s00208-004-0536-z

Cited by 49 publications

(81 citation statements)

References 10 publications

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“…We do know that it a compact, homogeneous but non-symmetric, multiply-connected, infinite dimensional complex Riemannian space. It is a projective, strictly almost Kähler manifold, a coadjoint orbit, hence a symplectic coset space of the volume preserving diffeomorphism group [26]. It is also the base manifold of a circle bundle over Gr(C n+1 ), where the U (1) holonomy provides a Berry phase.…”

Section: Geometry Of Gr(c N+1 )mentioning

confidence: 99%

Modeling Time’s Arrow

Jejjala

Kavic

Minić

et al. 2012

Entropy

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Quantum gravity, the initial low entropy state of the Universe, and the problem of time are interlocking puzzles. In this article, we address the origin of the arrow of time from a cosmological perspective motivated by a novel approach to quantum gravitation. Our proposal is based on a quantum counterpart of the equivalence principle, a general covariance of the dynamical phase space. We discuss how the nonlinear dynamics of such a system provides a natural description for cosmological evolution in the early Universe. We also underscore connections between the proposed non-perturbative quantum gravity model and fundamental questions in non-equilibrium statistical physics.

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Section: Geometry Of Gr(c N+1 )mentioning

confidence: 99%

Modeling Time’s Arrow

Jejjala

Kavic

Minić

et al. 2012

Entropy

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“…The expression of v H via the trace of the second fundamental form without reference to the mean curvature appeared in [3], Proposition 3. For 4D this theorem was obtained in [6] and for higher dimensions in [4], where we refer to for the proof.…”

Section: The Vortex Filaments Membranes and Skew-mean-curvature Flowmentioning

confidence: 99%

“…Note that this symplectic structure can be thought of as the 'total' averaging of the symplectic structures in each normal space N p P to P . (The Marsden-Weinstein structure in higher dimensions was studied in [2,3].) Furthermore, define the Hamiltonian function on those membranes by taking their (n − 2)-volume:…”

Section: The Vortex Filaments Membranes and Skew-mean-curvature Flowmentioning

confidence: 99%

The Vortex Filament Equation in any Dimension

Khesin

2013

Procedia IUTAM

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We present the vortex filament (or localized induction approximation) equation in any dimension. For an arbitrary n ≥ 3 the evolution of vorticity supported on vortex membranes of codimension 2 in R n is described by the skew (or binormal) mean-curvature flow, which generalises to any dimension the classical binormal equation in R 3 . This paper is a brief summary of the results in The vortex filament (or binormal) equation is the evolution equationof an arc-length parametrized space curve γ(·, t) ⊂ R 3 , where γ := ∂γ/∂θ. For an arbitrary parametrisation the filament equation reads ∂ t γ = k · b, where k and b = t × n stand, respectively, for the curvature value and binormal unit vector of the curve γ at the corresponding point. This binormal equation is known to be Hamiltonian with the Hamiltonian function given by the length functional H(γ) = length(γ) = γ γ (θ) dθ and relative to the Marsden-Weinstein symplectic structure on non-parametrized oriented space curves in R 3 , see e.g. [2,5]. At a curve γ this symplectic structure iswhere V and W are two vector fields attached to the curve γ and regarded as variations of this curve, while the volume form μ is evaluated on the three vectors V, W . Equivalently, the Marsden-Weinstein symplectic structure can be defined by means of the operator J of almost complex structure on curves: any variation V is rotated by the *

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“…Другими словами, приведенное выше расширение представляет собой расширение посредством дифференцирования [13]- [15].…”

Section: имеем точную последовательность алгебр лиunclassified

Double extensions of Lie algebras of Kac - Moody type and applications to some Hamiltonian systems

Roger¹

2013

ТМФ

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Описаны некоторые алгебры Ли типа Каца-Муди, построены их двойные расширения -центральные и полученные с помощью дифференцирований; для некоторых случаев также построены соответствующие группы Ли. Более по-дробно рассматривается случай алгебры Ли унимодулярных векторных полей. Для построения обобщений некоторых бесконечномерных гамильтоновых си-стем, подобных магнитогидродинамическим, используется линейная пуассонова структура на их регулярных двойственных пространствах.Ключевые слова: унимодулярные векторные поля, расширения алгебр Ли, гидроди-намика, магнитогидродинамика, коприсоединенные орбиты алгебр Ли.

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Non-linear Grassmannians as coadjoint orbits

Cited by 49 publications

References 10 publications

Modeling Time’s Arrow

Modeling Time’s Arrow

The Vortex Filament Equation in any Dimension

Double extensions of Lie algebras of Kac - Moody type and applications to some Hamiltonian systems

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