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.

Normal forms for linear Hamiltonian vector fields commuting with the action of a compact Lie group

1993 ◽  
Vol 114 (2) ◽  
pp. 235-268 ◽  
Author(s):  
Ian Melbourne ◽  
Michael Dellnitz
Keyword(s):  
Lie Group ◽  
Vector Fields ◽  
Normal Forms ◽  
Hamiltonian Vector ◽  
Compact Lie Group ◽  
The Real ◽  
Hamiltonian Vector Fields ◽  
Symplectic Action ◽  
Symplectic Matrices ◽  
Real Complex

AbstractWe obtain normal forms for infinitesimally symplectic matrices (or linear Hamiltonian vector fields) that commute with the symplectic action of a compact Lie group of symmetries. In doing so we extend Williamson's theorem on normal forms when there is no symmetry present.Using standard representation-theoretic results the symmetry can be factored out and we reduce to finding normal forms over a real division ring. There are three real division rings consisting of the real, complex and quaternionic numbers. Of these, only the real case is covered in Williamson's original work.

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.

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.

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].

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