Double extensions of Lie algebras of Kac-Moody type and applications to some hamiltonian systems (original) (raw)

Extended symmetries and Poisson algebras associated to twisted Dirac structures

Contemporary Mathematics, 2012

In this paper we study the relationship between the extended symmetries of exact Courant algebroids over a manifold M defined in [1] and the Poisson algebras of admissible functions associated to twisted Dirac structures when acted by Lie groups. We show that the usual homomorphisms of Lie algebras between the algebras of infinitesimal symmetries of the action, vector fields on the manifold and the Poisson algebra of observables, appearing in symplectic geometry, generalize to natural maps of Leibniz algebras induced both by the extended action and compatible moment maps associated to it in the context of twisted Dirac structures.

Poisson Algebras and 3D Superintegrable Hamiltonian Systems

Symmetry, Integrability and Geometry: Methods and Applications, 2018

Using a Poisson bracket representation, in 3D, of the Lie algebra sl(2), we first use highest weight representations to embed this into larger Lie algebras. These are then interpreted as symmetry and conformal symmetry algebras of the "kinetic energy", related to the quadratic Casimir function. We then consider the potentials which can be added, whilst remaining integrable, leading to families of separable systems, depending upon arbitrary functions of a single variable. Adding further integrals, in the superintegrable case, restricts these functions to specific forms, depending upon a finite number of arbitrary parameters. The Poisson algebras of these superintegrable systems are studied. The automorphisms of the symmetry algebra of the kinetic energy are extended to the full Poisson algebra, enabling us to build the full set of Poisson relations.

1+1) Schrodinger Lie bialgebras and their Poisson-Lie groups

1999

All Lie bialgebra structures for the (1+1)-dimensional centrally extended Schrodinger algebra are explicitly derived and proved to be of the coboundary type. Therefore, since all of them come from a classical r-matrix, the complete family of Schrodinger Poisson-Lie groups can be deduced by means of the Sklyanin bracket. All possible embeddings of the harmonic oscillator, extended Galilei and gl(2) Lie bialgebras within the Schrodinger classification are studied. As an application, new quantum (Hopf algebra) deformations of the Schrodinger algebra, including their corresponding quantum universal R-matrices, are constructed.

Poisson structures due to Lie algebra representations

International Journal of Theoretical Physics, 1995

Using a unitary solution of the classical Yang-Baxter equation on a Lie algebraG we describe a particular way of constructing homogeneous quadratic Poisson structures on the dual of aG-moduleV and study some local features of the symplectic foliation due to the involutive distribution of the Hamiltonian vector fields. We also give some examples where the symplectic leaves are explicitly calculated.

Double Lie algebras, semidirect product, and integrable systems

We study integrable systems on double Lie algebras in absence of Ad-invariant bilinear form by passing to the semidirect product with the tau\tau tau-representation. We show that in this stage a natural Ad-invariant bilinear form does exist, allowing for a straightforward application of the AKS theory, and giving rise to Manin triple structure, thus bringing the problem to the realm of Lie bialgebras and Poisson-Lie groups.

Infinitesimal Poisson algebras and linearization of Hamiltonian systems

Annals of Global Analysis and Geometry, 2020

Using the notion of a contravariant derivative, we give some algebraic and geometric characterizations of Poisson algebras associated to the infinitesimal data of Poisson submanifolds. We show that such a class of Poisson algebras provides a suitable framework for the study of the Hamiltonization problem for the linearized dynamics along Poisson submanifolds.

Lie–Hamilton systems on the plane: Properties, classification and applications

Journal of Differential Equations, 2015

We study Lie-Hamilton systems on the plane, i.e. systems of first-order differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of planar Hamiltonian vector fields with respect to a Poisson structure. We start with the local classification of finite-dimensional real Lie algebras of vector fields on the plane obtained in [A. González-López, N. Kamran and P.J. Olver, Proc. London Math. Soc. 64, 339 (1992)] and we interpret their results as a local classification of Lie systems. Moreover, by determining which of these real Lie algebras consist of Hamiltonian vector fields with respect to a Poisson structure, we provide the complete local classification of Lie-Hamilton systems on the plane. We present and study through our results new Lie-Hamilton systems of interest which are used to investigate relevant non-autonomous differential equations, e.g. we get explicit local diffeomorphisms between such systems. In particular, the Milne-Pinney, second-order Kummer-Schwarz, complex Riccati and Buchdahl equations as well as some Lotka-Volterra and nonlinear biomathematical models are analysed from this Lie-Hamilton approach.

Lie–Hamilton Systems: Theory and Applications

International Journal of Geometric Methods in Modern Physics, 2013

This work concerns the definition and analysis of a new class of Lie systems on Poisson manifolds enjoying rich geometric features: the Lie–Hamilton systems. We devise methods to study their superposition rules, time independent constants of motion and Lie symmetries, linearizability conditions, etc. Our results are illustrated by examples of physical and mathematical interest.

Classification and Casimir invariants of Lie–Poisson brackets

Physica D: Nonlinear Phenomena, 2000

We classify Lie-Poisson brackets that are formed from Lie algebra extensions. The problem is relevant because many physical systems owe their Hamiltonian structure to such brackets. A classification involves reducing all brackets to a set of normal forms, and is achieved partially through the use of Lie algebra cohomology. For extensions of order less than five, the number of normal forms is small and they involve no free parameters. We derive a general method of finding Casimir invariants of Lie-Poisson bracket extensions. The Casimir invariants of all low-order brackets are explicitly computed. We treat in detail a four field model of compressible reduced magnetohydrodynamics.