Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Poisson structures on complex flag manifolds associated with real forms

Authors: Philip Foth and Jiang-Hua Lu
Journal: Trans. Amer. Math. Soc. 358 (2006), 1705-1714
MSC (2000): Primary 53D17; Secondary 14M15, 22E15
Published electronically: September 22, 2005
MathSciNet review: 2186993
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a complex semisimple Lie group $G$ and a real form $G_0$ we define a Poisson structure on the variety of Borel subgroups of $G$ with the property that all $G_0$-orbits in $X$ as well as all Bruhat cells (for a suitable choice of a Borel subgroup of $G$) are Poisson submanifolds. In particular, we show that every non-empty intersection of a $G_0$-orbit and a Bruhat cell is a regular Poisson manifold, and we compute the dimension of its symplectic leaves.

References [Enhancements On Off] (What's this?)

  • 1. N. Andruskiewitsch and P. Jancsa. On simple real Lie bialgebras. Int. Math. Res. Not., 2004 (3), 139-158.MR 2038773
  • 2. J. Adams and D. Vogan. L-groups, projective representations, and the Langlands classification. Amer. J. Math., 114(1):45-138, 1991.MR 1147719 (93c:22021)
  • 3. S. Evens and J.-H. Lu. On the variety of Lagrangian subalgebras, I. Ann. Scient. Éc. Norm. Sup., 34: 631-668, 2001. MR 1862022 (2002i:17014)
  • 4. S. Evens and J.-H. Lu. Poisson harmonic forms, Kostant harmonic forms, and the $S^1$-equivariant cohomology of $K/T$. Adv. Math., 142:171-220, 1999. MR 1680047 (2001e:53085)
  • 5. S. Helgason. Differential geometry, Lie groups, and symmetric spaces. Pure and Applied Math., 80, Academic Press, 1978. MR 0514561 (80k:53081)
  • 6. A. W. Knapp. Lie Groups beyond an Introduction. Second Edition. Progress in Math., 140, Birkhäuser, Boston, 2002. MR 1920389 (2003c:22001)
  • 7. B. Kostant, Lie algebra cohomology and generalized Schubert cells, Ann. of Math., 77 (1) (1963), 72-144. MR 0142697 (26:266)
  • 8. B. Kostant and S. Kumar. The nil Hecke ring and cohomology of $G/P$ for a Kac-Moody group $G$. Adv. Math., 62(3):187-237, 1986. MR 0866159 (88b:17025b)
  • 9. J.-H. Lu. Poisson homogeneous spaces and Lie algebroids associated to Poisson actions. Duke Math. J., 86(2):261-304, 1997. MR 1430434 (98d:58204)
  • 10. J.-H. Lu. Coordinates on Schubert cells, Kostant's harmonic forms, and the Bruhat Poisson structure on $G/B$. Transform. Groups, 4:355-374, 1999.MR 1726697 (2001k:22018)
  • 11. J.-H. Lu and A. Weinstein. Poisson Lie groups, dressing transformations, and Bruhat decompositions. J. Diff. Geom., 31:501-526, 1990.MR 1037412 (91c:22012)
  • 12. R. W. Richardson and T. A. Springer. Combinatorics and geometry of $K$-orbits on flag manifolds. Contemporary Mathematics, Vol. 153, 109-142, 1993.MR 1247501 (94m:14065)
  • 13. Ya. Soibelman. Algebra of functions on a compact quantum group and its representations. Leningrad Math. J. 2:161-178, 1991. MR 1049910 (91i:58053a)
  • 14. J. A. Wolf. The action of a real semisimple Lie group on a complex flag manifold, I: Orbit structure and holomorphic arc components. Bull. Amer. Math. Soc., 75:1121-1237, 1969. MR 0251246 (40:4477)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53D17, 14M15, 22E15

Retrieve articles in all journals with MSC (2000): 53D17, 14M15, 22E15

Additional Information

Philip Foth
Affiliation: Department of Mathematics, University of Arizona, Tucson, Arizona 85721-0089

Jiang-Hua Lu
Affiliation: Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong

Keywords: Lie groups, real forms, flag varieties, Poisson structures, symplectic leaves
Received by editor(s): September 30, 2003
Received by editor(s) in revised form: June 16, 2004
Published electronically: September 22, 2005
Dedicated: Dedicated to Alan Weinstein on the occasion of his 60th birthday
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society