scholarly journals Odd Jacobi Manifolds and Loday-Poisson Brackets

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Andrew James Bruce
Keyword(s):  
Vector Field ◽  
Poisson Bracket ◽  
Vector Fields ◽  
Leibniz Rule ◽  
Poisson Brackets ◽  
Smooth Functions ◽  
Hamiltonian Vector Fields ◽  
Jacobi Manifolds ◽  
Symmetric Version ◽  
Jacobi Structure

We construct a nonskew symmetric version of a Poisson bracket on the algebra of smooth functions on an odd Jacobi supermanifold. We refer to such Poisson-like brackets as Loday-Poisson brackets. We examine the relations between the Hamiltonian vector fields with respect to both the odd Jacobi structure and the Loday-Poisson structure. Furthermore, we show that the Loday-Poisson bracket satisfies the Leibniz rule over the noncommutative product derived from the homological vector field.

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 Nambu–Poisson bracket on superspace

2018 ◽  
Vol 15 (11) ◽  
pp. 1850190 ◽  
Author(s):  
Viktor Abramov
Keyword(s):  
Poisson Bracket ◽  
Jacobi Identity ◽  
Leibniz Rule ◽  
Smooth Functions ◽  
Poisson Algebras ◽  
Fundamental Identity ◽  
Odd Degree ◽  
New Formula ◽  
Usual Formula

We propose an extension of [Formula: see text]-ary Nambu–Poisson bracket to superspace [Formula: see text] and construct by means of superdeterminant a family of Nambu–Poisson algebras of even degree functions, where the parameter of this family is an invertible transformation of Grassmann coordinates in superspace [Formula: see text]. We prove in the case of the superspaces [Formula: see text] and [Formula: see text] that our [Formula: see text]-ary bracket, defined with the help of superdeterminant, satisfies the conditions for [Formula: see text]-ary Nambu–Poisson bracket, i.e. it is totally skew-symmetric and it satisfies the Leibniz rule and the Filippov–Jacobi identity (fundamental identity). We study the structure of [Formula: see text]-ary bracket defined with the help of superdeterminant in the case of superspace [Formula: see text] and show that it is the sum of usual [Formula: see text]-ary Nambu–Poisson bracket and a new [Formula: see text]-ary bracket, which we call [Formula: see text]-bracket, where [Formula: see text] is the product of two odd degree smooth functions.

scholarly journals Generalization of Nambu–Hamilton Equation and Extension of Nambu–Poisson Bracket to Superspace

Universe ◽  
2018 ◽  
Vol 4 (10) ◽  
pp. 106 ◽  
Author(s):  
Viktor Abramov
Keyword(s):  
Vector Field ◽  
Poisson Bracket ◽  
Second Order ◽  
Leibniz Rule ◽  
Hamilton Equation ◽  
Fundamental Identity ◽  
Infinite Dimensional ◽  
Dimensional Group ◽  
The Right ◽  
Functional Matrix

We propose a generalization of the Nambu–Hamilton equation in superspace R 3 | 2 with three real and two Grassmann coordinates. We construct the even degree vector field in the superspace R 3 | 2 by means of the right-hand sides of the proposed generalization of the Nambu–Hamilton equation and show that this vector field is divergenceless in superspace. Then we show that our generalization of the Nambu–Hamilton equation in superspace leads to a family of ternary brackets of even degree functions defined with the help of a Berezinian. This family of ternary brackets is parametrized by the infinite dimensional group of invertible second order matrices, whose entries are differentiable functions on the space R 3 . We study the structure of the ternary bracket in a more general case of a superspace R n | 2 with n real and two Grassmann coordinates and show that for any invertible second order functional matrix it splits into the sum of two ternary brackets, where one is the usual Nambu–Poisson bracket, extended in a natural way to even degree functions in a superspace R n | 2 , and the second is a new ternary bracket, which we call the Ψ -bracket, where Ψ can be identified with an invertible second order functional matrix. We prove that the ternary Ψ -bracket as well as the whole ternary bracket (the sum of the Ψ -bracket with the usual Nambu–Poisson bracket) is totally skew-symmetric, and satisfies the Leibniz rule and the Filippov–Jacobi identity ( Fundamental Identity).

scholarly journals ∞-jets of diffeomorphisms preserving orbits of vector fields

2009 ◽  
Vol 7 (2) ◽  
Author(s):  
Sergiy Maksymenko
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Hamiltonian Vector ◽  
Homogeneous Polynomials ◽  
Local Flow ◽  
Linear Vector ◽  
Identity Component ◽  
Hamiltonian Vector Fields

AbstractLet F be a C ∞ vector field defined near the origin O ∈ ℝn, F(O) = 0, and (Ft) be its local flow. Denote by the set of germs of orbit preserving diffeomorphisms h: ℝn → ℝn at O, and let , (r ≥ 0), be the identity component of with respect to the weak Whitney Wr topology. Then contains a subset consisting of maps of the form Fα(x)(x), where α: ℝn → ℝ runs over the space of all smooth germs at O. It was proved earlier by the author that if F is a linear vector field, then = .In this paper we present a class of examples of vector fields with degenerate singularities at O for which formally coincides with , i.e. on the level of ∞-jets at O.We also establish parameter rigidity of linear vector fields and “reduced” Hamiltonian vector fields of real homogeneous polynomials in two variables.

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 On Symmetric Brackets Induced by Linear Connections

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1003
Author(s):  
Bogdan Balcerzak
Keyword(s):  
Vector Fields ◽  
Symplectic Manifolds ◽  
Smooth Functions ◽  
Hamiltonian Vector Fields ◽  
Linear Connections ◽  
Lie Bracket ◽  
Hermitian Manifolds ◽  
Almost Hermitian Manifolds ◽  
Hermitian Connection ◽  
Symplectic Connections

In this note, we discuss symmetric brackets on skew-symmetric algebroids associated with metric or symplectic structures. Given a pseudo-Riemannian metric structure, we describe the symmetric brackets induced by connections with totally skew-symmetric torsion in the language of Lie derivatives and differentials of functions. We formulate a generalization of the fundamental theorem of Riemannian geometry. In particular, we obtain an explicit formula of the Levi-Civita connection. We also present some symmetric brackets on almost Hermitian manifolds and discuss the first canonical Hermitian connection. Given a symplectic structure, we describe symplectic connections using symmetric brackets. We define a symmetric bracket of smooth functions on skew-symmetric algebroids with the metric structure and show that it has properties analogous to the Lie bracket of Hamiltonian vector fields on symplectic manifolds.

scholarly journals Generalization of Nambu-Hamilton Equation and Extension of Nambu-Poisson Bracket to Superspace

2018 ◽  
Author(s):  
Viktor Abramov
Keyword(s):  
Vector Field ◽  
Poisson Bracket ◽  
Second Order ◽  
Leibniz Rule ◽  
Hamilton Equation ◽  
Fundamental Identity ◽  
Infinite Dimensional ◽  
Dimensional Group ◽  
The Right ◽  
Functional Matrix

We propose a generalization of Nambu-Hamilton equation in superspace $\mathbb R^{3|2}$ with three real and two Grassmann coordinates. We construct the even degree vector field in the superspace $\mathbb R^{3|2}$ by means of the right-hand sides of proposed generalization of Nambu-Hamilton equation and show that this vector field is divergenceless in superspace. Then we show that our generalization of Nambu-Hamilton equation in superspace leads to family of ternary brackets of even degree functions defined with the help of Berezinian. This family of ternary brackets is parametrized by the infinite dimensional group of invertible second order matrices, whose entries are differentiable functions on the space $\mathbb R^{3}$. We study the structure of ternary bracket in a more general case of a superspace $\mathbb R^{n|2}$ with $n$ real and two Grassmann coordinates and show that for any invertible second order functional matrix it splits into the sum of two ternary brackets, where one is usual Nambu-Poisson bracket, extended in a natural way to even degree functions in a superspace $\mathbb R^{n|2}$, and the second is a new ternary bracket, which we call $\Psi$-bracket, where $\Psi$ can be identified with invertible second order functional matrix. We prove that ternary $\Psi$-bracket as well as the whole ternary bracket (the sum of $\Psi$-bracket with usual Nambu-Poisson bracket) is totally skew-symmetric, satisfies the Leibniz rule and the Filippov-Jacobi identity (Fundamental Identity).

Weak Center and Bifurcation of Critical Periods in a Cubic Z2-Equivariant Hamiltonian Vector Field

2015 ◽  
Vol 25 (11) ◽  
pp. 1550143 ◽  
Author(s):  
Yusen Wu ◽  
Wentao Huang ◽  
Yongqiang Suo
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Hamiltonian Vector ◽  
Local Bifurcation ◽  
Hamiltonian Vector Field ◽  
Critical Periods ◽  
Weak Center ◽  
Hamiltonian Vector Fields ◽  
Center Conditions ◽  
Bifurcation Of Critical Periods

This paper focuses on the problems of weak center and local bifurcation of critical periods for a class of cubic Z2-equivariant planar Hamiltonian vector fields. By computing the period constants carefully, one can see that there are three weak centers: (±1, 0) and the origin. The corresponding weak center conditions are also derived. Meanwhile, we address the problem of the coexistence of bifurcation of critical periods that occurred from (±1, 0) and the origin.

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.

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