It is known that symmetric orbits in g∗ for any simple Lie algebra g are equiped with a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the reduced Sklyanin bracket associated to the ”canonical” R-matrix. We realize quantization of this Poisson pencil on CPn type orbits (i.e. orbits in sl(n + 1)∗ whose real compact form is CPn) by means of q-deformed Verma modules. AMS Math...

We demonstrate that in a certain gauge the Lax matrices of the rational and hyperbolic Ruijsenaars–Schneider models have a quadratic r-matrix Poisson bracket which is an exact quadratization of the linear r–matrix Poisson bracket of the Calogero–Moser models. This phenomenon is explained by a geometric derivation of Lax equations for arbitrary flows of both hierarchies, which turn out to be gov...

The Poisson, contact and Nambu brackets define algebraic structures on C∞(M) satisfying the Jacobi identity or its generalization. The automorphism groups of these brackets are the symplectic, contact and volume preserving diffeomorphism groups. We introduce a modification of the Nambu bracket, which define an evolution equation generating the whole diffeomorphism group. The relation between th...

We study the general form of the * –commutator treated as a deformation of the Poisson bracket on the Grassman algebra. We show that, up to a similarity transformation , there are other deformations of the Poisson bracket in addition to the Moyal commutator (one at even and one at odd n, n is the number of the generators of the Grassman algebra) which are not reduced to the Moyal commutator by ...

The space P of pure states of any physical system, classical or quantum, is identified as a Poisson space with a transition probability. The latter is a function p : P × P → [0, 1]; in addition, a Poisson bracket is defined for functions on P . These two structures are connected through unitarity. Classical and quantum mechanics are each characterized by a simple axiom on the transition probabi...

We formulate the dynamical theory of nematic polymers, starting from a microscopic Poisson bracket approach. We find that the Poisson bracket between the nematic director and momentum depends on the (Maier-Saupe) order parameter of the nematic phase. We use this to derive reactive couplings of the nematic director to the strain rates. Additionally, we find that local dynamics breaks down as the...

Non commutative superspaces can be introduced as the MoyalWeyl quantization of a Poisson bracket for classical superfields. Different deformations are studied corresponding to constant background fields in string theory. Supersymmetric and non supersymmetric deformations can be defined, depending on the differential operators used to define the Poisson bracket. Some examples of deformed, 4 dime...

On the basis of the non-commutative q-calculus, we investigate a q-deformation of the classical Poisson bracket in order to formulate a generalized q-deformed dynamics in the classical regime. The obtained q-deformed Poisson bracket appears invariant under the action of the q-symplectic group of transformations. In this framework we introduce the q-deformed Hamilton’s equations and we derive th...

Abstract Let ???? be a finite-dimensional Lie algebra. The symmetric algebra (????) is equipped with the standard Lie–Poisson bracket. In this paper, we elaborate on surprising observation that one naturally associates second compatible Poisson bracket to any finite order automorphism ? of ????. We study related Poisson-commutative subalgebras (????; ?) ????(????) and associated contractions To...

This paper studies the dynamical fluid plus rigid-body system consisting of a two-dimensional rigid cylinder of general cross-sectional shape interacting with N point vortices. We derive the equations of motion for this system and show that, in particular, if the vortex strengths sum to zero and the rigid-body has a circular shape, the equations are Hamiltonian with respect to a Poisson bracket...