We study the transverse Poisson structure to adjoint orbits in a complex semi-simple Lie algebra. The problem is first reduced to the case of nilpotent orbits. We prove then that in suitably chosen quasi-homogeneous coordinates the quasi-degree of the transverse Poisson structure is −2. In the particular case of subregular nilpotent orbits we show that the structure may be computed by means of ...

In this paper we show that the moduli space of twisted polygons in G/P , where G is semisimple and P parabolic, and where g has two coordinated gradations, has a natural Poisson bracket that is directly linked to G-invariant evolutions of polygons. This structure is obtained by reducing the quotient twisted bracket on GN defined in [23] to the moduli space GN/PN . We prove that any Hamiltonian ...

The symplectic and Poisson structures on reduced phase spaces are reviewed, including the symplectic structure on coadjoint orbits of a Lie group and the Lie-Poisson structure on the dual of a Lie algebra. These results are applied to plasma physics. We show in three steps how the Maxwell-Vlasov equations for a collisionless plasma can be written in Hamiltonian form relative to a certain Poisso...

This paper considers random balls in a D-dimensional Euclidean space whose centers are prescribed by a homogeneous Poisson point process and whose radii are prescribed by a specific power law. A random field is constructed by counting the number of covering balls at each point. Even though it is not Gaussian, this field shares the same covariance function as the fractional Brownian field (fBf)....

Using inverse subordinators and Mittag-Leffler functions, we present a new definition of a fractional Poisson process parametrized by points of the Euclidean space R+. Some properties are given and, in particular, we prove a long-range dependence property.

Cubic interactions are considered in 3 and 7 space dimensions, respectively, for bosonic membranes in Poisson Bracket form. Their symmetries and vacuum configurations are discussed. Their associated first order equations are transformed to Nahm’s equations, and are hence seen to be integrable, for the 3-dimensional case, by virtue of the explicit Lax pair provided. The constructions introduced ...

A method for the deformation quantization of coadjoint orbits of semisimple Lie groups is proposed. It is based on the algebraic structure of the orbit. Its relation to geometric quantization and differentiable deformations is explored. Let G be a complex Lie group of dimension n and GR a real form of G. Let G and GR be their respective Lie algebras with Lie bracket [ , ]. As it is well known, ...

For any Poisson manifold P , the Poisson bracket on C∞(P ) extends to a Lie bracket on the space Ω(P ) of all differential one-forms, under which the space Z(P ) of closed one-forms and the space B(P ) of exact one-forms are Lie subalgebras. These Lie algebras are related by the exact sequence: 0 −→ R −→ C∞(P ) d −→ Z(P ) f −→ H(P,R) −→ 0, where H(P,R) is considered as a trivial Lie algebra, an...

We discuss in this paper the canonical structure of classical field theory in finite dimensions within the pataplectic Hamiltonian formulation, where we put forward the role of Legendre correspondance. We define the generalized Poisson p-brackets which are the analogues of the Poisson bracket on forms. We formulate the equations of motion of forms in terms of p-brackets. As illustration of our ...

In this paper we study geometric Poisson brackets and we show that, if M = (G n IRn)/G endowed with an affine geometry (in the Klein sense), and if G is a classical Lie group, then the geometric Poisson bracket for parametrized curves is a trivial extension of the one for unparametrized curves, except for the case G = GL(n, IR). This trivial extension does not exist in other nonaffine cases (pr...