scholarly journals Universal Central Extension of the Lie Algebra of Hamiltonian Vector Fields: Table 1.

2015 ◽  
Vol 2016 (16) ◽  
pp. 4996-5047 ◽  
Author(s):  
Bas Janssens ◽  
Cornelia Vizman
Keyword(s):  
Lie Algebra ◽  
Central Extension ◽  
Vector Fields ◽  
Hamiltonian Vector ◽  
Universal Central Extension ◽  
Hamiltonian Vector Fields

scholarly journals A Lie connection between Hamiltonian and Lagrangian optics

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Alex J. Dragt
Keyword(s):  
Vector Field ◽  
Lie Algebra ◽  
Vector Fields ◽  
Geometrical Optics ◽  
Hamiltonian Vector ◽  
Hamiltonian Vector Field ◽  
Third Order ◽  
Hamiltonian Vector Fields ◽  
Uniform Medium ◽  
International Audience

International audience It is shown that there is a non-Hamiltonian vector field that provides a Lie algebraic connection between Hamiltonian and Lagrangian optics. With the aid of this connection, geometrical optics can be formulated in such a way that all aberrations are attributed to ray transformations occurring only at lens surfaces. That is, in this formulation there are no aberrations arising from simple transit in a uniform medium. The price to be paid for this formulation is that the Lie algebra of Hamiltonian vector fields must be enlarged to include certain non-Hamiltonian vector fields. It is shown that three such vector fields are required at the level of third-order aberrations, and sufficient machinery is developed to generalize these results to higher order.

scholarly journals INFINITESIMAL TAKESAKI DUALITY OF HAMILTONIAN VECTOR FIELDS ON A SYMPLECTIC MANIFOLD

2000 ◽  
Vol 12 (12) ◽  
pp. 1669-1688 ◽  
Author(s):  
KATSUNORI KAWAMURA
Keyword(s):  
Lie Algebra ◽  
Symplectic Manifold ◽  
Vector Fields ◽  
Type Theorem ◽  
Hamiltonian Vector ◽  
Crossed Product ◽  
Hamiltonian Vector Fields ◽  
Symplectic Action

For an infinitesimal symplectic action of a Lie algebra [Formula: see text] on a symplectic manifold, we construct an infinitesimal crossed product of Hamiltonian vector fields and [Formula: see text]. We obtain its second crossed product in the case where [Formula: see text] and obtain an infinitesimal version of a Takesaki duality type theorem.

POISSON-NIJENHUIS STRUCTURES AND SATO HIERARCHY

1991 ◽  
Vol 03 (04) ◽  
pp. 403-466 ◽  
Author(s):  
G. MAGNANO ◽  
F. MAGRI
Keyword(s):  
Finite Number ◽  
Lie Algebra ◽  
Vector Fields ◽  
Hamiltonian Structure ◽  
Differential Operators ◽  
Hamiltonian Vector ◽  
Poisson Brackets ◽  
Moody Algebra ◽  
Hamiltonian Vector Fields ◽  
Direct Sum

We show that the direct sum of n copies of a Lie algebra is endowed with a sequence of affine Lie-Poisson brackets, which are pairwise compatible and define a multi-Hamiltonian structure; to this structure one can associate a recursion operator and a Kac-Moody algebra of Hamiltonian vector fields. If the initial Lie algebra is taken to be an associative algebra of differential operators, a suitable family of Hamiltonian vector fields reproduce either the n-th Gel'fand-Dikii hierarchy (for n finite) or Sato's hierarchy (for n = ∞). Within the same framework, it is also possible to recover a class of integro-differential hierarchies involving a finite number of fields, which generalize the Gel'fand-Dikii equations and are equivalent to Sato's hierarchy.

scholarly journals Hamiltonian Vector Fields on Weil Bundles

2015 ◽  
Vol 7 (3) ◽  
Author(s):  
Norbert Mahoungou Moukala ◽  
Basile Guy Richard Bossoto
Keyword(s):  
Vector Fields ◽  
Hamiltonian Vector ◽  
Hamiltonian Vector Fields

A weight-homogenous condition to the real Jacobian conjecture in

2021 ◽  
pp. 1-9
Author(s):  
Francisco Braun ◽  
Claudia Valls
Keyword(s):  
Differential Equations ◽  
Vector Fields ◽  
Qualitative Theory ◽  
Hamiltonian Vector ◽  
Jacobian Conjecture ◽  
Local Diffeomorphism ◽  
The Real ◽  
Hamiltonian Vector Fields ◽  
Real Linear

Abstract It is known that a polynomial local diffeomorphism $(f,\, g): {\mathbb {R}}^{2} \to {\mathbb {R}}^{2}$ is a global diffeomorphism provided the higher homogeneous terms of $f f_x+g g_x$ and $f f_y+g g_y$ do not have real linear factors in common. Here, we give a weight-homogeneous framework of this result. Our approach uses qualitative theory of differential equations. In our reasoning, we obtain a result on polynomial Hamiltonian vector fields in the plane, generalization of a known fact.

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