Quadratic Poisson Brackets Research Articles

Quadratic Poisson brackets compatible with an algebra structure

Quadratic Poisson brackets on a vector space equipped with a bilinear multiplication are studied. A notion of a bracket compatible with the multiplication is introduced and an effective criterion of such compatibility is given. Among compatible brackets, a subclass of coboundary brackets is described, and such brackets are enumerated in a number of examples.

Quantum [formula omitted] in any dimension

The κ-deformation of the D-dimensional Poincaré algebra ( D ≥ 2) with any signature is given. Further the quadratic Poisson brackets, determined by the classical r-matrix are calculated, and a construction of the quantum Poincaré group “with noncommuting parameters” is proposed.

Quadratic brackets from symplectic forms

We give a physicist oriented survey of Poisson-Lie symmetries of classical systems. We consider finite-dimensional geometric actions and the chiral WZNW model as examples for the general construction. An essential point is the appearance of quadratic Poisson brackets for group-like variables. It is believed that upon quantization they lead to quadratic exchange algebras.

Hamiltonian structures of the Melnikov system and its reductions

The bi-Hamiltonian structure of an integrable dynamical system introduced by Melnikov (1986) is studied. This equation arises as a symmetry constraint of the KP hierarchy via squared eigenfunctions and can be understood as a Boussinesq system with a source. The standard linear and quadratic Poisson brackets associated with the space of pseudo-differential symbols are used to derive two compatible Hamiltonian operators. A bi-Hamiltonian formulation for the Drinfeld-Sokolov system is derived via reduction techniques.

On the bi-Hamiltonian structure of Toda and relativistic Toda lattices

We introduce a general quadratic Poisson bracket on the associative algebra equipped with non-degenerate scalar product. With the help of this bracket we obtain the interpretation of the Toda and relativistic Toda lattices as the restrictions of one and the same bi-Hamiltonian system to two different low-dimensional manifolds, which are Poisson submanifolds with respect to two brackets simultaneously.

Canonical charge algebras for integrable fermionic theories

The non-local charges for a class of chiral invariant fermionic models as well as for the Gross-Neveu models are shown to obey a quadratic Poisson bracket algebra. Both commuting and anticommuting spinors are considered. The classical r-matrices for these models are found. The Schwinger terms in the quantum current algebra are computed.