Quantum Groups can be constructed by applying the quantization by deformation procedure to Lie groups endowed with a suitable Poisson bracket. Here we try to develop an understanding of these structures by investigating dynamical systems which are associated with this bracket. We look at SU (2) and SU (1, 1), as submanifolds of a 4–dimensional phase space with constraints, and deal with two cla...

We extend the Poisson bracket from a Lie bracket of phase space functions to a Lie bracket of functions on the space of canonical histories and investigate the resulting algebras. Typically, such extensions define corresponding Lie algebras on the space of Lagrangian histories via pull back to a space of partial solutions. These are the same spaces of histories studied with regard to path integ...

A new Poisson bracket for Hamiltonian forms on the full multisymplectic phase space is defined. At least for forms of degree n − 1, where n is the dimension of space-time, Jacobi's identity is fulfilled.

In terms of a 3-Lie algebra and the classical Poisson bracket {Bn,L} dKP hierarchy, special 3-bracket {Bm,Bn,L} is proposed. When m = 0 or 1, 3-lax equation ∂L∂t={Bm,Bn,L} hierarchy corresponding proof given. Meanwhile, for generalized case (m,n), also investigated.

We develop a Hamiltonian discontinuous finite element discretization of a generalized Hamiltonian system for linear hyperbolic systems, which include the rotating shallow water equations, the acoustic and Maxwell equations. These equations have a Hamiltonian structure with a bilinear Poisson bracket, and as a consequence the phase-space structure, “mass” and energy are preserved. We discretize ...

The moduli space of Higgs bundles can be constructed as a quotient an infinite-dimensional and hence admits orbit type decomposition. In this paper, we show that the decomposition is complex Whitney stratification such each stratum symplectic submanifold Poisson bracket. Moreover, these brackets glue to bracket on structure sheaf so stratified space.

We describe all Poisson brackets compatible with the natural cluster algebra structure in the open Schubert cell of the Grassmannian Gk(n) and show that any such bracket endows Gk(n) with a structure of a Poisson homogeneous space with respect to the natural action of SLn equipped with an R-matrix Poisson-Lie structure. The corresponding R-matrices belong to the simplest class in the Belavin-Dr...

The Lie algebroid [10] is a generalization of both concepts of Lie algebra and integrable distribution, being a vector bundle (E, π, M) with a Lie bracket on his space of sections with properties very similar to those of a tangent bundle. The Poisson manifolds are the smooth manifolds equipped with a Poisson bracket on their ring of functions. I have to remark that the cotangent bundle of a Poi...