Abstract
We study the Poisson-Lie structures on the group SU(2,R). We calculate all Poisson-Lie structures on SU(2,R) through the correspondence with Lie bialgebra structures on its Lie algebra su(2,R). We show that all these structures are linearizable in the neighborhood of the unity of the group SU(2,R). Finally, we show that the Lie algebra consisting of all infinitesimal automorphisms is strictly contained in the Lie algebra consisting of Hamiltonian vector fields.
Highlights
A constant solution of mCYBE r on a given Lie algebra provide a coboundary Poisson-Lie structure π on group G given by π ( s=) rs* r − ls* r, ∀s ∈ G, (7)
Recall that for semisimple Lie algebras, all Lie bialgebra structures are coboundaries, and the corresponding Poisson-Lie structures can be fully solved through the classical r-matrices
Let us notice that the Lie bialgebra structure δ associated to π defines a linear Poisson-Lie structure on
Summary
Where lx* and ry* respectively denote the left and right translations in Poisson-Lie structure π has rank zero at a neutral element e of G , i.e., G by x and π (e) = 0 . Of neutral element e of G , the Poisson-Lie structure π reads (2014) On the Structure of Infinitesimal Automorphisms of the Poisson-Lie Group By this Poisson bracket, C∞ (G) becomes a Lie algebra.
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