# A classification of Poisson homogeneous spaces of complex reductive Poisson-Lie groups

Banach Center Publications (2000)

- Volume: 51, Issue: 1, page 103-108
- ISSN: 0137-6934

## Access Full Article

top## Abstract

top## How to cite

topKarolinsky, Eugene. "A classification of Poisson homogeneous spaces of complex reductive Poisson-Lie groups." Banach Center Publications 51.1 (2000): 103-108. <http://eudml.org/doc/209021>.

@article{Karolinsky2000,

abstract = {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.},

author = {Karolinsky, Eugene},

journal = {Banach Center Publications},

keywords = {Poisson homogeneous -spaces; Sklyanin bracket; Lagrangian subalgebras},

language = {eng},

number = {1},

pages = {103-108},

title = {A classification of Poisson homogeneous spaces of complex reductive Poisson-Lie groups},

url = {http://eudml.org/doc/209021},

volume = {51},

year = {2000},

}

TY - JOUR

AU - Karolinsky, Eugene

TI - A classification of Poisson homogeneous spaces of complex reductive Poisson-Lie groups

JO - Banach Center Publications

PY - 2000

VL - 51

IS - 1

SP - 103

EP - 108

AB - 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.

LA - eng

KW - Poisson homogeneous -spaces; Sklyanin bracket; Lagrangian subalgebras

UR - http://eudml.org/doc/209021

ER -

## References

top- [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). Zbl0553.58040
- [2] N. Bourbaki, Groupes et algèbres de Lie, ch. 4-6, Hermann, Paris, 1968.
- [3] V. G. Drinfeld, On Poisson homogeneous spaces of Poisson-Lie groups, Theor. Math. Phys. 95 (1993), 226-227.
- [4] V. G. Drinfeld, Quantum Groups, in: Proceedings of the International Congress of Mathematicians, 1986, Berkeley, 1987, 798-820.
- [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. Zbl0797.22001
- [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).
- [7] L.-C. Li and S. Parmentier, Nonlinear Poisson structures and r-matrices, Commun. Math. Phys. 125 (1989), 545-563. Zbl0695.58011
- [8] J.-H. Lu, Classical dynamical r-matrices and homogeneous Poisson structures on G/H and K/T, math. SG/9909004.
- [9] A. L. Onishchik and E. B. Vinberg, Lie groups and algebraic groups, Springer-Verlag, Berlin, 1990. Zbl0722.22004
- [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.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.