Une caractérisation du fibré cotangent

1988 ◽  
Vol 38 (3) ◽  
pp. 465-472 ◽  
Author(s):  
Tong van Duc
Keyword(s):  
Lie Algebra ◽  
Direct Product ◽  
Cotangent Bundle ◽  
Vector Fields ◽  
Total Space ◽  
Lie Derivation ◽  
Infinitesimal Automorphisms

We prove that the Lie algebra of infinitesimal automorphisms of the cotangent structure on the total space of the cotangent bundle of a manifold is isomorphic to the semi-direct product of the Lie algebra of the vector fields on the manifold by the space of closed 1-forms, the vector fields operating on the forms by Lie derivation. The derivations of this algebra Lie are completely determined and we prove that it characterises the cotangent bundle.

scholarly journals Deforming 𝔥-trivial the Lie algebra Vect(S1) inside the Lie algebra of pseudodifferential operators Ψ𝒟𝒪

2017 ◽  
Vol 14 (06) ◽  
pp. 1750082 ◽  
Author(s):  
Imed Basdouri ◽  
Issam Bartouli ◽  
Jean Lerbet
Keyword(s):  
Lie Algebra ◽  
Cotangent Bundle ◽  
Vector Fields ◽  
Pseudodifferential Operators ◽  
Sufficient Conditions ◽  
Lie Derivative ◽  
Smooth Vector ◽  
Algebra Of Functions ◽  
Infinitesimal Deformations ◽  
Necessary And Sufficient

In this paper, we consider the action of Vect(S1) by Lie derivative on the spaces of pseudodifferential operators [Formula: see text]. We study the [Formula: see text]-trivial deformations of the standard embedding of the Lie algebra Vect(S1) of smooth vector fields on the circle, into the Lie algebra of functions on the cotangent bundle [Formula: see text]. We classify the deformations of this action that become trivial once restricted to [Formula: see text], where [Formula: see text] or [Formula: see text]. Necessary and sufficient conditions for integrability of infinitesimal deformations are given.

scholarly journals Siegel domains over self-dual cones and their automorphisms

1974 ◽  
Vol 55 ◽  
pp. 33-80 ◽  
Author(s):  
Tadashi Tsuji
Keyword(s):  
Lie Algebra ◽  
Vector Fields ◽  
Polynomial Vector ◽  
Siegel Domain ◽  
Polynomial Vector Fields ◽  
Siegel Domains ◽  
Dual Cones ◽  
Graded Lie Algebra ◽  
Infinitesimal Automorphisms ◽  
Affine Part

The Lie algebra gr of all infinitesimal automorphisms of a Siegel domain in terms of polynomial vector fields was investigated by Kaup, Matsushima and Ochiai [6]. It was proved in [6] that gr is a graded Lie algebra; gr = g-1 + g-1/2 + g0 + g1/2 + g1 and the Lie subalgebra ga of all infinitesimal affine automorphisms is given by the graded subalgebra; ga = g-1 + g-1/2 + g0. Nakajima [9] proved without the assumption of homogeneity that the non-affine parts g1/2 and g1 can be determined from the affine part ga.

A note on the algebra of Poisson brackets

1984 ◽  
Vol 96 (1) ◽  
pp. 45-60 ◽  
Author(s):  
C. J. Atkin
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symplectic Manifold ◽  
Vector Fields ◽  
Poisson Brackets ◽  
Infinite Dimensional

In a long sequence of notes in the Comptes Rendus and elsewhere, and in the papers [1], [2], [3], [6], [7], Lichnerowicz and his collaborators have studied the ‘classical infinite-dimensional Lie algebras’, their derivations, automorphisms, co-homology, and other properties. The most familiar of these algebras is the Lie algebra of C∞ vector fields on a C∞ manifold. Another is the Lie algebra of ‘Poisson brackets’, that is, of C∞ functions on a C∞ symplectic manifold, with the Poisson bracket as composition; some questions concerning this algebra are of considerable interest in the theory of quantization – see, for instance, [2] and [3].

scholarly journals Representations of the Lie algebra of vector fields on a sphere

2019 ◽  
Vol 223 (8) ◽  
pp. 3581-3593 ◽  
Author(s):  
Yuly Billig ◽  
Jonathan Nilsson
Keyword(s):  
Lie Algebra ◽  
Vector Fields

scholarly journals SOME NOTES ON THE MODIFIED RIEMANNIAN EXTENSION g ̃_(∇,c) ON COTANGENT BUNDLE

2020 ◽  
Vol 20 (4) ◽  
pp. 931-940
Author(s):  
HASIM CAYIR
Keyword(s):  
Cotangent Bundle ◽  
Vector Fields ◽  
Lie Derivatives ◽  
Complete And Vertical Lifts

In this paper, we define the modified Riemannian extension g ̃_(∇,c) in the cotangent bundle T^* M, which is completely determined by its action on complete lifts of vector fields. Later, we obtain the covarient and Lie derivatives applied to the modified Riemannian extension with respect to the complete and vertical lifts of vector and kovector fields, respectively.

scholarly journals On Lie algebras of vector fields with invariant submanifolds

1974 ◽  
Vol 55 ◽  
pp. 91-110 ◽  
Author(s):  
Akira Koriyama
Keyword(s):  
Lie Algebras ◽  
Compact Support ◽  
Vector Fields ◽  
Finite Dimensional ◽  
Invariant Submanifolds ◽  
Infinitesimal Automorphisms

It is known (Pursell and Shanks [9]) that an isomorphism between Lie algebras of infinitesimal automorphisms of C∞ structures with compact support on manifolds M and M′ yields an isomorphism between C∞ structures of M and M’.Omori [5] proved that this is still true for some other structures on manifolds. More precisely, let M and M′ be Hausdorff and finite dimensional manifolds without boundary. Let α be one of the following structures:

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