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 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).

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.

scholarly journals NILPOTENT CLASSICAL MECHANICS

2007 ◽  
Vol 22 (14n15) ◽  
pp. 2513-2533 ◽  
Author(s):  
ANDRZEJ FRYDRYSZAK
Keyword(s):  
Poisson Bracket ◽  
Symplectic Geometry ◽  
Classical Mechanics ◽  
Leibniz Rule ◽  
Hamiltonian Form ◽  
Symmetric Form ◽  
Generalized Poisson ◽  
Generalized Differential ◽  
Special Case ◽  
R Symmetry

The formalism of nilpotent mechanics is introduced in the Lagrangian and Hamiltonian form. Systems are described using nilpotent, commuting coordinates η. Necessary geometrical notions and elements of generalized differential η-calculus are introduced. The so-called s-geometry, in a special case when it is orthogonally related to a traceless symmetric form, shows some resemblances to the symplectic geometry. As an example of an η-system the nilpotent oscillator is introduced and its supersymmetrization considered. It is shown that the R-symmetry known for the graded superfield oscillator also present here for the supersymmetric η-system. The generalized Poisson bracket for (η, p)-variables satisfies modified Leibniz rule and has nontrivial Jacobiator.

scholarly journals SYMMETRIES AND INVARIANTS OF TWISTED QUANTUM ALGEBRAS AND ASSOCIATED POISSON ALGEBRAS

2008 ◽  
Vol 20 (02) ◽  
pp. 173-198 ◽  
Author(s):  
A. I. MOLEV ◽  
E. RAGOUCY
Keyword(s):  
Poisson Bracket ◽  
Quantum Algebras ◽  
Poisson Algebras ◽  
Triangular Matrices ◽  
Polynomial Invariants ◽  
Enveloping Algebra ◽  
Quantized Enveloping Algebras ◽  
Quantized Enveloping Algebra ◽  
Group B ◽  
Mathematical Literature

We construct an action of the braid group BN on the twisted quantized enveloping algebra [Formula: see text] where the elements of BN act as automorphisms. In the classical limit q → 1, we recover the action of BN on the polynomial functions on the space of upper triangular matrices with ones on the diagonal. The action preserves the Poisson bracket on the space of polynomials which was introduced by Nelson and Regge in their study of quantum gravity and rediscovered in the mathematical literature. Furthermore, we construct a Poisson bracket on the space of polynomials associated with another twisted quantized enveloping algebra [Formula: see text]. We use the Casimir elements of both twisted quantized enveloping algebras to reproduce and construct some well-known and new polynomial invariants of the corresponding Poisson algebras.

scholarly journals Mutation-periodic quivers, integrable maps and associated Poisson algebras

2011 ◽  
Vol 369 (1939) ◽  
pp. 1264-1279 ◽  
Author(s):  
Allan P. Fordy
Keyword(s):  
Poisson Bracket ◽  
Canonical Transformation ◽  
Hamiltonian Structure ◽  
Poisson Algebra ◽  
Complete Integrability ◽  
Canonical Coordinates ◽  
Poisson Algebras ◽  
Invariant Functions ◽  
Liouville’S Equation

We consider a class of map, recently derived in the context of cluster mutation. In this paper, we start with a brief review of the quiver context, but then move onto a discussion of a related Poisson bracket, along with the Poisson algebra of a special family of functions associated with these maps. A bi-Hamiltonian structure is derived and used to construct a sequence of Poisson-commuting functions and hence show complete integrability. Canonical coordinates are derived, with the map now being a canonical transformation with a sequence of commuting invariant functions. Compatibility of a pair of these functions gives rise to Liouville’s equation and the map plays the role of a Bäcklund transformation.

scholarly journals Generalized derivations and general relativity

2013 ◽  
Vol 91 (10) ◽  
pp. 757-763
Author(s):  
Michael Heller ◽  
Tomasz Miller ◽  
Leszek Pysiak ◽  
Wiesław Sasin
Keyword(s):  
General Relativity ◽  
Extra Dimension ◽  
Linear Mapping ◽  
Einstein Equations ◽  
Leibniz Rule ◽  
Generalized Derivations ◽  
Field Equations ◽  
Time Dimension ◽  
Smooth Functions ◽  
Kaluza Klein

We construct differential geometry (connection, curvature, etc.) based on generalized derivations of an algebra [Formula: see text]. Such a derivation, introduced by Brešar in 1991, is given by a linear mapping [Formula: see text] such that there exists a usual derivation, d, of [Formula: see text] satisfying the generalized Leibniz rule u(ab) = u(a)b + ad(b) for all [Formula: see text]. The generalized geometry “is tested” in the case of the algebra of smooth functions on a manifold. We then apply this machinery to study generalized general relativity. We define the Einstein–Hilbert action and deduce from it Einstein’s field equations. We show that for a special class of metrics containing, besides the usual metric components, only one nonzero term, the action reduces to the O’Hanlon action that is the Brans–Dicke action with potential and with the parameter ω equal to zero. We also show that the generalized Einstein equations (with zero energy–stress tensor) are equivalent to those of the Kaluza–Klein theory satisfying a “modified cylinder condition” and having a noncompact extra dimension. This opens a possibility to consider Kaluza–Klein models with a noncompact extra dimension that remains invisible for a macroscopic observer. In our approach, this extra dimension is not an additional physical space–time dimension but appears because of the generalization of the derivation concept.

Z-Graded Extensions of Poisson Brackets

1997 ◽  
Vol 09 (01) ◽  
pp. 1-27 ◽  
Author(s):  
Janusz Grabowski
Keyword(s):  
Poisson Bracket ◽  
Jacobi Identity ◽  
Differential Forms ◽  
Poisson Manifold ◽  
Exterior Derivative ◽  
Tensor Fields ◽  
Lie Bracket ◽  
Nijenhuis Bracket ◽  
Hamiltonian Map ◽  
Vector Valued

A Z-graded Lie bracket { , }P on the exterior algebra Ω(M) of differential forms, which is an extension of the Poisson bracket of functions on a Poisson manifold (M,P), is found. This bracket is simultaneously graded skew-symmetric and satisfies the graded Jacobi identity. It is a kind of an 'integral' of the Koszul–Schouten bracket [ , ]P of differential forms in the sense that the exterior derivative is a bracket homomorphism: [dμ, dν]P=d{μ, ν}P. A naturally defined generalized Hamiltonian map is proved to be a homomorphism between { , }P and the Frölicher–Nijenhuis bracket of vector valued forms. Also relations of this graded Poisson bracket to the Schouten–Nijenhuis bracket and an extension of { , }P to a graded bracket on certain multivector fields, being an 'integral' of the Schouten–Nijenhuis bracket, are studied. All these constructions are generalized to tensor fields associated with an arbitrary Lie algebroid.

scholarly journals Four-dimensional gravity as an almost-Poisson system

2015 ◽  
Vol 24 (07) ◽  
pp. 1550047
Author(s):  
Eyo Eyo Ita
Keyword(s):  
Phase Space ◽  
Poisson Bracket ◽  
Jacobi Identity ◽  
Equations Of Motion ◽  
Space Structure ◽  
Lagrange Equations ◽  
Phase Space Structure ◽  
Full Phase Space ◽  
Dimensional Gravity ◽  
Definition Of

In this paper, we examine the phase space structure of a noncanonical formulation of four-dimensional gravity referred to as the Instanton representation of Plebanski gravity (IRPG). The typical Hamiltonian (symplectic) approach leads to an obstruction to the definition of a symplectic structure on the full phase space of the IRPG. We circumvent this obstruction, using the Lagrange equations of motion, to find the appropriate generalization of the Poisson bracket. It is shown that the IRPG does not support a Poisson bracket except on the vector constraint surface. Yet there exists a fundamental bilinear operation on its phase space which produces the correct equations of motion and induces the correct transformation properties of the basic fields. This bilinear operation is known as the almost-Poisson bracket, which fails to satisfy the Jacobi identity and in this case also the condition of antisymmetry. We place these results into the overall context of nonsymplectic systems.

scholarly journals Poisson Brackets and Structure of Nongraded Hamiltonian Lie Algebras Related to Locally-Finite Derivations

2003 ◽  
Vol 55 (4) ◽  
pp. 856-896 ◽  
Author(s):  
Yucai Su
Keyword(s):  
Poisson Bracket ◽  
Lie Algebras ◽  
Poisson Brackets ◽  
Cohomology Groups ◽  
Poisson Algebras ◽  
Locally Finite ◽  
Isomorphism Classes ◽  
Sandwich Method

AbstractXu introduced a class of nongraded Hamiltonian Lie algebras. These Lie algebras have a Poisson bracket structure. In this paper, the isomorphism classes of these Lie algebras are determined by employing a “sandwich” method and by studying some features of these Lie algebras. It is obtained that two Hamiltonian Lie algebras are isomorphic if and only if their corresponding Poisson algebras are isomorphic. Furthermore, the derivation algebras and the second cohomology groups are determined.

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