On the 12th of April 2018, at 10:15 a.m.
Javier de Lucas Araujo (KMMF WFUW)
will give a talk on
"Poisson—Hopf algebra deformations
of a class of Hamiltonian systems"
Abstract
We provide a method of deformation of a class of Hamiltonian systems via Poisson—Hopf algebras that allows for the algebraic derivation of their constants of the motion and the geometric description of their dynamical properties. More specifically, we start by a nonautonomous Hamiltonian system on a Poisson manifold N whose dynamic is determined by a tdependent Hamiltonian taking values in a finitedimensional Lie algebra of functions isomorphic to an abstract Lie algebra g. The Hamiltonian system is then attached to the universal enveloping algebra U(g) and the Poisson algebra C^{∞}(g^{*}) relative to the Kirillov—Kostant—Souriau bracket. The deformed quantum algebras U_{z}(g) and Poisson—Hopf algebras C_{z}^{∞}(g^{*}), along with the induced symplectic foliations on g^{*} for each z, allow for the deformation of the tdependent Hamiltonian function of our original system. This originates a zparametrized family of Hamiltonian systems on N whose constants of the motion can be derived through Casimir elements of U_{z}(g) and C_{z}^{∞}(g^{*}). The coalgebra structure of C_{z}^{∞}(g^{*}) enables us to derive multidimensional generalizations of the original Hamiltonian system. Among other applications, we deform a onedimensional Winternitz—Smorodisnky oscillator to obtain a family of oscillators with a positiondependent mass and we provide their generalizations to other higherdimensional manifolds.
The seminar takes place on Thursdays from 10:15 a.m. to 12:00
in the room 2.23 of the main building of the Faculty of Physics (the 2nd floor), Pasteur Str. 5,
Warszawa.
Additional information can be found on the webpage
http://oldwww.fuw.edu.pl/KMMF/sem.czw.przedp.html.
