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1.
A locally convex Lie algebra is said to be locally exponential if it belongs to some local Lie group in canonical coordinates. In this note we give criteria
for locally exponential Lie algebras of vector fields on an infinite-dimensional manifold to integrate to global Lie group
actions. Moreover, we show that all necessary conditions are satisfied if the manifold is finite-dimensional connected and
σ-compact, which leads to a generalization of Palais’ Integrability Theorem.
相似文献
2.
On the Construction of Geometric Integrators in the RKMK Class 总被引:2,自引:0,他引:2
Kenth Engø 《BIT Numerical Mathematics》2000,40(1):41-61
We consider the construction of geometric integrators in the class of RKMK methods. Any differential equation in the form of an infinitesimal generator on a homogeneous space is shown to be locally equivalent to a differential equation on the Lie algebra corresponding to the Lie group acting on the homogeneous space. This way we obtain a distinction between the coordinate-free phrasing of the differential equation and the local coordinates used. In this paper we study methods based on arbitrary local coordinates on the Lie group manifold. By choosing the coordinates to be canonical coordinates of the first kind we obtain the original method of Munthe-Kaas [16]. Methods similar to the RKMK method are developed based on the different coordinatizations of the Lie group manifold, given by the Cayley transform, diagonal Padé approximants of the exponential map, canonical coordinates of the second kind, etc. Some numerical experiments are also given. 相似文献
3.
We construct the class of integrable classical and quantum systems on the Hopf algebras describing n interacting particles.
We obtain the general structure of an integrable Hamiltonian system for the Hopf algebra A(g) of a simple Lie algebra g and
prove that the integrals of motion depend only on linear combinations of k coordinates of the phase space, 2·ind g≤k≤g·ind g, whereind g andg are the respective index and Coxeter number of the Lie algebra g. The standard procedure of q-deformation results in the
quantum integrable system. We apply this general scheme to the algebras sl(2), sl(3), and o(3, 1). An exact solution for the
quantum analogue of the N-dimensional Hamiltonian system on the Hopf algebra A(sl(2)) is constructed using the method of noncommutative
integration of linear differential equations.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 373–390, September, 2000 相似文献
4.
Marte Rørvik Høyem 《Acta Appl Math》2010,109(1):61-73
We consider the Lie algebra that corresponds to the Lie pseudogroup of all conformal transformations on the plane. This conformal
Lie algebra is canonically represented as the Lie algebra of holomorphic vector fields in ℝ2≃ℂ. We describe all representations of
\mathfrakg\mathfrak{g}
via vector fields in J
0ℝ2=ℝ3(x,y,u), which project to the canonical representation, and find their algebra of scalar differential invariants. 相似文献
5.
Summary. We present a new class of integration methods for differential equations on manifolds, in the framework of Lie group actions.
Canonical coordinates of the second kind is used for representing the Lie group locally by means of its corresponding Lie
algebra. The coordinate map itself can, in many cases, be computed inexpensively, but the approach also involves the inversion
of its differential, a task that can be challenging. To succeed, it is necessary to consider carefully how to choose a basis
for the Lie algebra, and the ordering of the basis is important as well. For semisimple Lie algebras, one may take advantage
of the root space decomposition to provide a basis with desirable properties. The problem of ordering leads us to introduce
the concept of an admissible ordered basis (AOB). The existence of an AOB is established for some of the most important Lie
algebras. The computational cost analysis shows that the approach may lead to more efficient solvers for ODEs on manifolds
than those based on canonical coordinates of the first kind presented by Munthe-Kaas. Numerical experiments verify the derived
properties of the new methods.
Received April 2, 1999 / Revised version received January 18, 2000 / Published online August 2, 2000 相似文献
6.
Classical and quantum mechanics based on an extended Heisenberg algebra with additional canonical commutation relations for
position and momentum coordinates are considered. In this approach additional noncommutativity is removed from the algebra
by a linear transformation of coordinates and transferred to the Hamiltonian (Lagrangian). This linear transformation does
not change the quadratic form of the Hamiltonian (Lagrangian), and Feynman’s path integral preserves its exact expression
for quadratic models. The compact general formalism presented here can be easily illustrated in any particular quadratic case.
As an important result of phenomenological interest, we give the path integral for a charged particle in the noncommutative
plane with a perpendicular magnetic field. We also present an effective Planck constant ħ
eff which depends on additional noncommutativity. 相似文献
7.
On any manifold, any nondegenerate symmetric 2-form (metric) and any nondegenerate skew-symmetric differential form ω can
be reduced to a canonical form at any point but not in any neighborhood: the corresponding obstructions are the Riemannian
tensor and dω. The obstructions to flatness (to reducibility to a canonical form) are well known for any G-structure, not
only for Riemannian or almost symplectic structures. For a manifold with a nonholonomic structure (nonintegrable distribution),
the general notions of flatness and obstructions to it, although of huge interest (e.g., in supergravity) were not known until
recently, although particular cases have been known for more than a century (e.g., any contact structure is nonholonomically
“flat”: it can always be reduced locally to a canonical form). We give a general definition of the nonholonomic analogues
of the Riemann tensor and its conformally invariant analogue, the Weyl tensor, in terms of Lie algebra cohomology and quote
Premet’s theorems describing these cohomologies. Using Premet’s theorems and the SuperLie package, we calculate the tensors
for flag manifolds associated with each maximal parabolic subalgebra of each simple Lie algebra (and in several more cases)
and also compute the obstructions to flatness of the G(2)-structure and its nonholonomic superanalogue.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 2, pp. 186–219, November, 2007. 相似文献
8.
Invariant approach to optimal investment–consumption problem: the constant elasticity of variance (CEV) model
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Ahmet Bakkaloglu Taha Aziz Aeeman Fatima F.M. Mahomed Chaudry Masood Khalique 《Mathematical Methods in the Applied Sciences》2017,40(5):1382-1395
The optimal investment–consumption problem under the constant elasticity of variance (CEV) model is solved using the invariant approach. Firstly, the invariance criteria for scalar linear second‐order parabolic partial differential equations in two independent variables are reviewed. The criteria is then employed to reduce the CEV model to one of the four Lie canonical forms. It is found that the invariance criteria help in transforming the original equation to the second Lie canonical form and with a proper parameter selection; the required transformation converts the original equation to the first Lie canonical form that is the heat equation. As a consequence, we find some new classes of closed‐form solutions of the CEV model for the case of reduction into heat equation and also into second Lie canonical form. The closed‐form analytical solution of the Cauchy initial value problems for the CEV model under investigation is also obtained. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
9.
We construct torus bundles over locally symmetric varieties associated to cocycles in the cohomology group , where Γ is a discrete subgroup of a semisimple Lie group and L is a lattice in a real vector space. We prove that such a torus bundle has a canonical complex structure and that the space
of holomorphic forms of the highest degree on a fiber product of such bundles is isomorphic to the space of mixed automorphic
forms of a certain type.
(Received 4 September 1998) 相似文献
10.
The Clebsch method provides a unifying approach for deriving variational principles for continuous and discrete dynamical
systems where elements of a vector space are used to control dynamics on the cotangent bundle of a Lie group via a velocity
map. This paper proves a reduction theorem which states that the canonical variables on the Lie group can be eliminated, if
and only if the velocity map is a Lie algebra action, thereby producing the Euler–Poincaré (EP) equation for the vector space
variables. In this case, the map from the canonical variables on the Lie group to the vector space is the standard momentum
map defined using the diamond operator. We apply the Clebsch method in examples of the rotating rigid body and the incompressible
Euler equations. Along the way, we explain how singular solutions of the EP equation for the diffeomorphism group (EPDiff)
arise as momentum maps in the Clebsch approach. In the case of finite-dimensional Lie groups, the Clebsch variational principle
is discretized to produce a variational integrator for the dynamical system. We obtain a discrete map from which the variables
on the cotangent bundle of a Lie group may be eliminated to produce a discrete EP equation for elements of the vector space.
We give an integrator for the rotating rigid body as an example. We also briefly discuss how to discretize infinite-dimensional
Clebsch systems, so as to produce conservative numerical methods for fluid dynamics.
相似文献
11.
We analyze the structure of the reduced phase space that arises in the Hamiltonian reduction of the phase space of free particle
motion over the groupSL(2, ℝ). The reduction considered is based on introducing constraints that are analogous to those used in the reduction of the Wess-Zumino-Novikov-Witten
model to Toda systems. It is shown that the reduced phase space is diffeomorphic either to a union of two two-dimensional
planes or to a cylinder S1×ℝ. We construct canonical coordinates for both cases and show that in the first case, the reduced phase space is symplectomorphic
to the union of two cotangent bundles T*(ℝ) endowed with a canonical symplectic structure, while in the second case, it is symplectomorphic to the cotangent bundle T*
(S1), which is also endowed with a canonical symplectic structure.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 1, pp. 149–161, January, 1997. 相似文献
12.
Niels Vigand Pedersen 《Aequationes Mathematicae》1994,48(2-3):228-253
Summary For a simply connected solvable Lie group we specify the structure of the product, the inverse and the exponential map expressed in suitable coordinates (canonical coordinates of the second kind), and point out that in these coordinates the product and inverse are expressed entirely in terms of polynomials, exponential functions and trigonometric functions. We devise algorithms for computing the product, the inverse and the exponential map. 相似文献
13.
A. A. Magazev I. V. Shirokov Yu. A. Yurevich 《Theoretical and Mathematical Physics》2008,156(2):1127-1141
On Lie group manifolds, we consider right-invariant magnetic geodesic flows associated with 2-cocycles of the corresponding Lie algebras. We investigate the algebra of the integrals of motion of magnetic geodesic flows
and also formulate a necessary and sufficient condition for their integrability in quadratures, giving the canonical forms
of 2-cocycles for all four-dimensional Lie algebras and selecting integrable cases.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 189–206, August, 2008. 相似文献
14.
We offer a method for constructing invariants of the coadjoint representation of Lie groups that reduces this problem to known problems of linear algebra. This method is based on passing to symplectic coordinates on the coadjoint representation orbits, which play the role of local coordinates on those orbits. The corresponding transition functions are their parametric equations. Eliminating the symplectic coordinates from the transition functions, we can obtain the complete set of invariants. The proposed method allows solving the problem of constructing invariants of the coadjoint representation for Lie groups with an arbitrary dimension and structure. 相似文献
15.
In this paper, we prove that the natural metric on the connected component of the unit in the (Lie) motion group of a compact
Finsler manifold supplied with its inner metric generates a bi-invariant inner Finsler metric. The latter is defined by the
invariant Chebyshev norm on the Lie algebra of generators of 1-parameter motion subgroups on the manifold. This norm is equal
to the maximal value of the generator’s length. A δ-homogeneous manifold is characterized by the condition that the canonical
projection of the component onto the manifold is a submetry with respect to their inner metrics. The Chebyshev norms for the
Euclidean spheres, the Berger spheres, and homogeneous Riemannian metrics on the 3-dimensional complex projective space are
found. This gives interesting examples of invariant norms on Lie algebras and a new method for the separating of delta-homogeneous
but not normal metrics.
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 60, Algebra,
2008. 相似文献
16.
Clestin Wafo Soh 《Communications in Nonlinear Science & Numerical Simulation》2010,15(2):139-143
We show that the structure of the Lie symmetry algebra of a system of n linear second-order ordinary differential equations with constant coefficients depends on at most n-1 parameters. The tools used are Jordan canonical forms and appropriate scaling transformations. We put our approach to test by presenting a simple proof of the fact that the dimension of the symmetry Lie algebra of a system of two linear second-order ordinary differential with constant coefficients is either 7, 8 or 15. Also, we establish for the first time that the dimension of the symmetry Lie algebra of a system of three linear second-order ordinary differential equations with constant coefficients is 10, 12, 13 or 24. 相似文献
17.
We obtain necessary and sufficient conditions for the integrability in quadratures of geodesic flows on homogeneous spaces M with invariant and central metrics. The proposed integration algorithm consists in using a special canonical transformation in the space T
*
M based on constructing the canonical coordinates on the orbits of the coadjoint representation and on the simplectic sheets of the Poisson algebra of invariant functions. This algorithm is applicable to integrating geodesic flows on homogeneous spaces of a wild Lie group. 相似文献
18.
Block introduced certain analogues of the Zassenhaus algebras over a field of characteristic 0. The nongraded infinite-dimensional simple Lie algebras of Block type constructed by Xu can be viewed as generalizations of the Block algebras. In this paper, we construct a family of irreducible modules in terms of multiplication and differentiation operators on "polynomials" for four-devivation nongraded Lie algebras of Block type based on the finite-dimensional irreducible weight modules with multiplicity one of general linear Lie algebras. We also find a new series of submodules from which some irreducible quotient modules are obtained. 相似文献
19.
20.
Baranovskii S. P. Mikheev V. V. Shirokov I. V. 《Theoretical and Mathematical Physics》2001,129(1):1311-1319
We consider equations on Lie groups and classical and quantum Hamiltonian systems on coadjoint representation orbits. We show that the transition to canonical coordinates on orbits of the coadjoint representation allows constructing semiclassical solutions and the corresponding spectra of quantum equations such that all the symmetries of the original problem are preserved. Our method is used to find the semiclassical spectrum of the asymmetric quantum top. 相似文献