Some applications of the odd Poisson bracket to the description of the classical and quantum dynamics are represented.

In symplectic geometry, it is often useful to consider the so-called Poisson bracket on the algebra of functions on a C ∞ symplectic manifold M. The bracket determines, and is determined by, the symplectic form; however, many of the features of symplectic geometry are more conveniently described in terms of the Poisson bracket. When one turns to the study of symplectic manifolds in the holomorp...

We exhibit a Poisson module restoring a twisted Poincaré duality between Poisson homology and cohomology for the polynomial algebra R = C[X1, . . . , Xn] endowed with Poisson bracket arising from a uniparametrised quantum affine space. This Poisson module is obtained as the semiclassical limit of the dualising bimodule for Hochschild homology of the corresponding quantum affine space. As a coro...

Following an earlier work we derive a gauge-independent canonical structure for a fully relativistic multicomponent plasma theory. The Klimontovich form of the distribution function is used to derive the basic Poisson bracket relations for the canonical variables/~, B, and E. The Poisson bracket relations provide an explicit canonical realization of the Lie algebra of the Poincar6 group and the...

The double bracket dissipation approach is applied to the Vlasov kinetic equation. The Vlasov equation is then transformed by a Poisson map to moment dynamics, leading to a nonlocal form of Darcy’s law. Next, kinetic equations for particles with anisotropic interaction are considered and also cast into the double bracket dissipation form. The moment dynamics for these double bracket kinetic equ...

Let M be a manifold with an action of a Lie group G, A the function algebra on M . The first problem we consider is to construct a Uh(g) invariant quantization, Ah, of A, where Uh(g) is a quantum group corresponding to G. Let s be a G invariant Poisson bracket on M . The second problem we consider is to construct a Uh(g) invariant two parameter (double) quantization, At,h, of A such that At,0 i...

The widely accepted approach to the foundation of quantum mechanics is that the Poisson bracket, governing the non-commutative algebra of operators, is taken as a postulate with no underlying physics. In this manuscript, it is shown that this postulation is in fact unnecessary and may be replaced by a few deeper concepts, which ultimately lead to the derivation of Poisson bracket. One would onl...

New realizations of observables in dynamical systems with second class constraints. Abstract In the Dirac bracket approach to dynamical systems with second class constraints observables are represented by elements of a quotient Dirac bracket algebra. We describe families of new realizations of this algebra through quotients of the original Poisson algebra. Explicite expressions for generators a...

When X is an associative H-space, the bar spectral sequence computes the homology of the delooping, H∗(BX). If X is an n-fold loop space for n ≥ 2 this is a spectral sequence of Hopf algebras. Using machinery by Sugawara and Clark, we show that the spectral sequence filtration respects the Browder bracket structure on H∗(BX), and so it is moreover a spectral sequence of Poisson algebras. Throug...