共查询到20条相似文献,搜索用时 15 毫秒
1.
Pavol Ševera 《Letters in Mathematical Physics》2015,105(12):1689-1701
2.
We explicitly construct a class of coboundary Poisson–Lie structures on the group of formal diffeomorphisms of
n
. Equivalently, these give rise to a class of coboundary triangular Lie bialgebra structures on the Lie algebra W
n
of formal vector fields on
n
. We conjecture that this class accounts for all such coboundary structures. The natural action of the constructed Poisson–Lie diffeomorphism groups gives rise to large classes of compatible Poisson structures on
n
, thus making it a Poisson space. Moreover, the canonical action of the Poisson–Lie groups FDiff(
m
) × FDiff
n
) gives rise to classes of compatible Poisson structures on the space J
(
m
,
n
) of infinite jets of smooth maps
m
n
, which makes it also a Poisson space for this action. Poisson modules of generalized densities are also constructed. Initial steps towards a classification of these structures are taken. 相似文献
3.
4.
In this letter, first we give a decomposition for any Lie–Poisson structure associated to the modular vector. In particular, splits into two compatible Lie–Poisson structures if . As an application, we classified quadratic deformations of Lie– Poisson structures on up to linear diffeomorphisms.
Research partially supported by NSF of China and the Research Project of “Nonlinear Science”. 相似文献
5.
The trigonometric Ruijsenaars–Schneider model is derived by symplectic reduction of Poisson–Lie symmetric free motion on the group U(n). The commuting flows of the model are effortlessly obtained by reducing canonical free flows on the Heisenberg double of U(n). The free flows are associated with a very simple Lax matrix, which is shown to yield the Ruijsenaars–Schneider Lax matrix upon reduction. 相似文献
6.
Poisson–Lie target space duality is a framework where duality transformations are properly defined. In this Letter, we investigate the dual pair of -models defined by the double SO(3,1) in the Iwasawa decomposition. 相似文献
7.
We present the proof of the one loop renormalizability in the strict field theoretic sense of the Poisson–Lie σ-models. The result is valid for any Drinfeld double and it relies solely on the Poisson–Lie structure encoded in the target manifold. 相似文献
8.
We prove that on the Sobolev spaces H
N
(S
1) (N≥ 0), each leaf of the foliation, induced by the second Poisson bracket of KdV, admits global action-angle variables. The
actions with respect to the first bracket raise to the actions with respect to the second bracket. The angles for the first
bracket are, at the same time, angles for the second bracket.
Received: 12 November 1999 / Accepted: 9 May 2000 相似文献
9.
《Physics letters. A》2001,291(6):389-396
The relation between dissipation and the symplectic structure of the momentum-space is studied in so(3) Lie algebra and in 2D fluid dynamics. Three kinds of dissipative mechanisms are discussed and a general bracket formalism is introduced. A chaotic dynamical system due to Lorenz, and largely studied in low-dimensional models of geophysical fluid dynamics, is analysed in its geometric and dynamical features, by means of the formalism previously introduced. A mechanism of energy transfer for this low-order model is discussed. 相似文献
10.
The Lie–Rinehart algebra of a (connected) manifold ${\mathcal {M}}$ , defined by the Lie structure of the vector fields, their action and their module structure over ${C^\infty({\mathcal {M}})}$ , is a common, diffeomorphism invariant, algebra for both classical and quantum mechanics. Its (noncommutative) Poisson universal enveloping algebra ${\Lambda_{R}({\mathcal {M}})}$ , with the Lie–Rinehart product identified with the symmetric product, contains a central variable (a central sequence for non-compact ${{\mathcal {M}}}$ ) ${Z}$ which relates the commutators to the Lie products. Classical and quantum mechanics are its only factorial realizations, corresponding to Z = i z, z = 0 and ${z = \hbar}$ , respectively; canonical quantization uniquely follows from such a general geometrical structure. For ${z =\hbar \neq 0}$ , the regular factorial Hilbert space representations of ${\Lambda_{R}({\mathcal{M}})}$ describe quantum mechanics on ${{\mathcal {M}}}$ . For z = 0, if Diff( ${{\mathcal {M}}}$ ) is unitarily implemented, they are unitarily equivalent, up to multiplicity, to the representation defined by classical mechanics on ${{\mathcal {M}}}$ . 相似文献
11.
12.
Dirk Blömker Stanislaus Maier-Paape Thomas Wanner 《Communications in Mathematical Physics》2001,223(3):553-582
This paper gives theoretical results on spinodal decomposition for the stochastic Cahn–Hilliard–Cook equation, which is a
Cahn–Hilliard equation perturbed by additive stochastic noise. We prove that most realizations of the solution which start
at a homogeneous state in the spinodal interval exhibit phase separation, leading to the formation of complex patterns of
a characteristic size.
In more detail, our results can be summarized as follows. The Cahn–Hilliard–Cook equation depends on a small positive parameter
ε which models atomic scale interaction length. We quantify the behavior of solutions as ε→ 0. Specifically, we show that
for the solution starting at a homogeneous state the probability of staying near a finite-dimensional subspace ?ε is high as long as the solution stays within distance r
ε=O(ε
R
) of the homogeneous state. The subspace ?ε is an affine space corresponding to the highly unstable directions for the linearized deterministic equation. The exponent
R depends on both the strength and the regularity of the noise.
Received: 2 May 2000 / Accepted: 8 July 2001 相似文献
13.
《Journal of Nonlinear Mathematical Physics》2013,20(1):90-93
Abstract Operators of non–local symmetry are used to construct exact solutions of nonlinear heat equations. 相似文献
14.
For a Lie algebra with Lie bracket got by taking commutators in a nonunital associative algebra
, let
be the vector space of tensors over
equipped with the Itô Hopf algebra structure derived from the associative multiplication in
. It is shown that a necessary and sufficient condition that the double product integral
satisfy the quantum Yang–Baxter equation over
is that
satisfy the same equation over the unital associative algebra
got by adjoining a unit element to
. In particular, the first-order coefficient r1 of r[h] satisfies the classical Yang–Baxter equation. Using the fact that the multiplicative inverse of
is
where
is the inverse of
in
we construct a quantisation of an arbitrary quasitriangular Lie bialgebra structure on
in the unital associative subalgebra of
consisting of formal power series whose zero order coefficient lies in the space
of symmetric tensors. The deformation coproduct acts on
by conjugating the undeformed coproduct by
and the coboundary structure r of
is given by
where
is the flip.Mathematical Subject Classification (2000). 53D55, 17B62 相似文献
15.
We use a generalized Ricci tensor, defined for generalized metrics in Courant algebroids, to show that Poisson–Lie T-duality is compatible with the 1-loop renormalization group flow. 相似文献
16.
We develop a method of asymptotic study of the integrated density of states (IDS) N(E) of a random Schr?dinger operator with a non-positive (attractive) Poisson potential. The method is based on the periodic
approximations of the potential instead of the Dirichlet-Neumann bracketing used before. This allows us to derive more precise
bounds for the rate of approximations of the IDS by the IDS of respective periodic operators and to obtain rigorously for
the first time the leading term of log N(E) as E→−∞ for the Poisson random potential with a singular single-site (impurity) potential, in particular, for the screened Coulomb
impurities, dislocations, etc.
Received: 18 November 1998 / Accepted: 9 March 1999 相似文献
17.
N. P. Landsman 《Communications in Mathematical Physics》2001,222(1):97-16
It is well known that a measured groupoid G defines a von Neumann algebra W
*(G), and that a Lie groupoid G canonically defines both a C
*-algebra C
*(G) and a Poisson manifold A
*(G). We construct suitable categories of measured groupoids, Lie groupoids, von Neumann algebras, C
*-algebras, and Poisson manifolds, with the feature that in each case Morita equivalence comes down to isomorphism of objects.
Subsequently, we show that the maps G↦W
*(G), G↦C
*(G), and G↦A
*(G) are functorial between the categories in question. It follows that these maps preserve Morita equivalence.
Received: 6 December 2000 / Accepted: 19 April 2001 相似文献
18.
Cornelia Vizman 《Letters in Mathematical Physics》2008,84(2-3):245-256
We present an abstract Kelvin–Noether theorem for geodesic equations on abelian Lie group extensions with right invariant metrics and we apply it to equations of hydrodynamical type. Another Kelvin–Noether theorem for a class of central extensions of semidirect products is shown. 相似文献
19.
Denis Perrot 《Communications in Mathematical Physics》2001,218(2):373-391
We consider a smooth groupoid of the form Σ⋊Γ, where Σ is a Riemann surface and Γ a discrete pseudogroup acting on Σ by local
conformal diffeomorphisms. After defining a K-cycle on the crossed product C
0(Σ)⋊Γ generalising the classical Dolbeault complex, we compute its Chern character in cyclic cohomology, using the index theorem
of Connes and Moscovici. This involves in particular a generalisation of the Euler class constructed from the modular automorphism
group of the von Neumann algebra L
∞(Σ)⋊Γ.
Received: 1 February 2000 / Accepted: 3 December 2000 相似文献
20.
Fraydoun Rezakhanlou 《Communications in Mathematical Physics》2000,211(2):413-438
We study the asymptotic behavior of , where u solves the Hamilton–Jacobi equation u
t
+H(x,u
x
) ≡ 0 with H a stationary ergodic process in the x-variable. It was shown in Rezakhanlou–Tarver [RT] that u
ɛ converges to a deterministic function provided H(x,p) is convex in p and the convex conjugate of H in the p-variable satisfies certain growth conditions. In this article we establish a central limit theorem for the convergence by
showing that for a class of examples, u
ɛ(x,t) can be (stochastically) represented as
, where Z(x,t) is a suitable random field. In particular we establish a central limit theorem when the dimension is one and , where ω is a random function that enjoys some mild regularity.
Received: 15 February 1999 / Accepted: 14 December 1999 相似文献