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.
References
[1]
Drinfeld’s, V.G. (1983) Hamiltonian Structures on Lie Groups, Lie Bialgebras and the Geometric Meaning of the Classical Yang-Baxter Equations. Soviet Mathematics—Doklady, 27, 68-71.
[2]
Drinfeld, V.G. (1986) Quantum Groups, Proceedings of the International Congress of Mathematicians, Berkley, 3-11 August 1986, 789-820.
[3]
Lu, J.H. and Weinstein, A. (1990) Poisson-Lie Group, Dressing Transformaions and Bruhat Decomposition. Journal of Differential Geometry, 31, 301-599.
[4]
Semenove-Tian-Shasky, M.A. (1983) What Is a Classical r-Matrix. Functional Analysis and Its Applications, 17, 259-272. http://dx.doi.org/10.1007/BF01076717
[5]
Chari, V. and Pressley, A. (1994) A Guide to Quantum Groups. Cambridge University Press, Cambridge.
[6]
Belavin, A.A. and Drinfeld, V.G. (1983) Solution of the Classical Yang-Baxter Equation for Simple Lie Algebras. Functional Analysis and Its Applications, 16, 159-180. http://dx.doi.org/10.1007/BF01081585
[7]
Chloup-Arnould, V. (1997) Linearization of Some Poisson-Lie Tensor. Journal of Geometry and Physics, 24, 145-195.
[8]
Dufour, J.P. (1990) Linarisation de certaines structures de Poisson. Journal of Differential Geometry, 32, 415-428.
[9]
Vaisman, I. (1990) Remarks on the Lichnerowicz-Poisson Cohomology. Annales de l’Institut Fourier, 40, 951-963. http://dx.doi.org/10.5802/aif.1243
[10]
Conn, J. (1984) Normal Forms for Analytic Poisson Structures. Annals of Mathematics, 119, 576-601. http://dx.doi.org/10.2307/2007086
[11]
Conn, J. (1985) Normal Forms for Smooth Poisson Structures. Annals of Mathematics, 121, 565-593. http://dx.doi.org/10.2307/1971210
