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Title

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

Authors 1

Affiliations

  1. Department of Mathematics and Mechanics, Kharkov State University, 4 Svobody Sq., Kharkov, 310077, Ukraine

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 D (here = Lie G). A geometric interpretation of some Poisson homogeneous G-spaces is also proposed.

Bibliography

  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).
  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.
  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.
  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.
  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.
Banach Center Publications
2000/51/1
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Pages:
103-108
Main language of publication
English
Published
2000
Exact and natural sciences