Let G be a complex reductive connected algebraic group equipped with the Sklyanin bracket. A classification of Poisson homogeneous G-spaces with connected isotropy subgroups is given. This result is based on Drinfeld's correspondence between Poisson homogeneous G-spaces and Lagrangian subalgebras in the double D𝖌 (here 𝖌 = Lie G). A geometric interpretation of some Poisson homogeneous G-spaces is also proposed.
@article{bwmeta1.element.bwnjournal-article-bcpv51z1p103bwm,
author = {Karolinsky, Eugene},
title = {A classification of Poisson homogeneous spaces of complex reductive Poisson-Lie groups},
journal = {Banach Center Publications},
volume = {51},
year = {2000},
pages = {103-108},
zbl = {0981.53078},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv51z1p103bwm}
}
Karolinsky, Eugene. A classification of Poisson homogeneous spaces of complex reductive Poisson-Lie groups. Banach Center Publications, Tome 51 (2000) pp. 103-108. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv51z1p103bwm/
[000] [1] A. A. Belavin and V. G. Drinfeld, Triangle equations and simple Lie algebras, in: Soviet Scientific Reviews, Section C 4, 1984, 93-165 (2nd edition: Classic Reviews in Mathematics and Mathematical Physics 1, Harwood, Amsterdam, 1998). | Zbl 0553.58040
[001] [2] N. Bourbaki, Groupes et algèbres de Lie, ch. 4-6, Hermann, Paris, 1968.
[002] [3] V. G. Drinfeld, On Poisson homogeneous spaces of Poisson-Lie groups, Theor. Math. Phys. 95 (1993), 226-227.
[003] [4] V. G. Drinfeld, Quantum Groups, in: Proceedings of the International Congress of Mathematicians, 1986, Berkeley, 1987, 798-820.
[004] [5] V. V. Gorbatsevich, A. L. Onishchik and E. B. Vinberg, Structure of Lie groups and Lie algebras, Encyclopaedia of Math. Sci. 41, Springer-Verlag, Berlin, 1994. | Zbl 0797.22001
[005] [6] E. A. Karolinsky, A classification of Poisson homogeneous spaces of a compact Poisson-Lie group, Mathematical Physics, Analysis, and Geometry 3 (1996), 274-289 (in Russian).
[006] [7] L.-C. Li and S. Parmentier, Nonlinear Poisson structures and r-matrices, Commun. Math. Phys. 125 (1989), 545-563. | Zbl 0695.58011
[007] [8] J.-H. Lu, Classical dynamical r-matrices and homogeneous Poisson structures on G/H and K/T, math. SG/9909004.
[008] [9] A. L. Onishchik and E. B. Vinberg, Lie groups and algebraic groups, Springer-Verlag, Berlin, 1990. | Zbl 0722.22004
[009] [10] S. Parmentier, Twisted affine Poisson structures, decomposition of Lie algebras, and the Classical Yang-Baxter equation, preprint MPI/91-82, Max-Planck-Institut für Mathematik, Bonn, 1991.