Lie bi-algebra structures for centrally extended two-dimensional Galilei algebra and their Lie-Poisson counterparts

Abstract

All bialgebra structures for centrally extended Galilei algebra are classified. The corresponding Lie-Poisson structures on centrally extended Galilei group are found.

Canonical, Lie-Algebraic and Quadratic Twist Deformations of Galilei Group

“…First of all, one can consider basic dynamical models corresponding to considered Galilei algebras [55]. Besides, it seems interesting to find other D = 3 + 1 dimensional nonrelativistic space-times, corresponding Hopf algebras describing symmetry, and dual groups, predicted by the general classification of all Galilean Poisson-Lie structures [32] (see also for two-dimensional case [56] and [57]). One can also ask about a "superposition" of discussed quantum deformations as well as their supersymmetric N = 1 extensions (see e.g.…”

Poisson Structures on the Galilei Group

“…The method we use (c.f. [2]) is based on solving directly the cocycle condition; it has been already used for finding Poisson-Lie structures on the two-dimensional Galilei group [3], [4].…”

Classification of low-dimensional Lie super-bialgebras

“…Two-dimensional Galilei algebra admits 9 inequivalent Lie bialgebra structures and only one out of them is a coboundary [13]. The central extension of Galilei algebra by the mass operator admits 26 inequivalent Lie bialgebra structures out of which 8 are coboundaries [14]. Similar analysis aimed mainly on the construction of quantum groups was performed before in the cases of Heisenberg-Weyl and oscillator Lie algebras [15].…”