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

Eugene Karolinsky

Banach Center Publications (2000)

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

Abstract

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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.

How to cite

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Karolinsky, 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

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  1. [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. [2] N. Bourbaki, Groupes et algèbres de Lie, ch. 4-6, Hermann, Paris, 1968. 
  3. [3] V. G. Drinfeld, On Poisson homogeneous spaces of Poisson-Lie groups, Theor. Math. Phys. 95 (1993), 226-227. 
  4. [4] V. G. Drinfeld, Quantum Groups, in: Proceedings of the International Congress of Mathematicians, 1986, Berkeley, 1987, 798-820. 
  5. [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. [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. [7] L.-C. Li and S. Parmentier, Nonlinear Poisson structures and r-matrices, Commun. Math. Phys. 125 (1989), 545-563. Zbl0695.58011
  8. [8] J.-H. Lu, Classical dynamical r-matrices and homogeneous Poisson structures on G/H and K/T, math. SG/9909004. 
  9. [9] A. L. Onishchik and E. B. Vinberg, Lie groups and algebraic groups, Springer-Verlag, Berlin, 1990. Zbl0722.22004
  10. [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. 

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