On Homogeneous and Symmetric CR Manifolds

Andrea Altomani; Costantino Medori; Mauro Nacinovich

Bollettino dell'Unione Matematica Italiana (2010)

  • Volume: 3, Issue: 2, page 221-265
  • ISSN: 0392-4041

Abstract

top
We consider canonical fibrations and algebraic geometric structures on homogeneous CR manifolds, in connection with the notion of CR algebra. We give applications to the classifications of left invariant CR structures on semisimple Lie groups and of CR-symmetric structures on complete flag varieties.

How to cite

top

Altomani, Andrea, Medori, Costantino, and Nacinovich, Mauro. "On Homogeneous and Symmetric CR Manifolds." Bollettino dell'Unione Matematica Italiana 3.2 (2010): 221-265. <http://eudml.org/doc/290681>.

@article{Altomani2010,
abstract = {We consider canonical fibrations and algebraic geometric structures on homogeneous CR manifolds, in connection with the notion of CR algebra. We give applications to the classifications of left invariant CR structures on semisimple Lie groups and of CR-symmetric structures on complete flag varieties.},
author = {Altomani, Andrea, Medori, Costantino, Nacinovich, Mauro},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {221-265},
publisher = {Unione Matematica Italiana},
title = {On Homogeneous and Symmetric CR Manifolds},
url = {http://eudml.org/doc/290681},
volume = {3},
year = {2010},
}

TY - JOUR
AU - Altomani, Andrea
AU - Medori, Costantino
AU - Nacinovich, Mauro
TI - On Homogeneous and Symmetric CR Manifolds
JO - Bollettino dell'Unione Matematica Italiana
DA - 2010/6//
PB - Unione Matematica Italiana
VL - 3
IS - 2
SP - 221
EP - 265
AB - We consider canonical fibrations and algebraic geometric structures on homogeneous CR manifolds, in connection with the notion of CR algebra. We give applications to the classifications of left invariant CR structures on semisimple Lie groups and of CR-symmetric structures on complete flag varieties.
LA - eng
UR - http://eudml.org/doc/290681
ER -

References

top
  1. ALTOMANI, A. - MEDORI, C. - NACINOVICH, M., The CR structure of minimal orbits in complex flag manifolds, J. Lie Theory, 16, no. 3 (2006), 483-530. Zbl1120.32023MR2248142
  2. ALTOMANI, A. - MEDORI, C. - NACINOVICH, M., Orbits of real forms in complex flag manifolds, Ann. Scuola Norm. Sup. Pisa. Cl. Sci. (5) vol. IX (2010), 1-41. MR2668874
  3. ANDREOTTI, A. - FREDRICKS, G. A., Embeddability of real analytic Cauchy-Riemann manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 6, no. 2 (1979), 285-304. Zbl0449.32008MR541450
  4. AZAD, H. - HUCKLEBERRY, A. - RICHTHOFER, W., Homogeneous CR-manifolds, J. Reine Angew. Math., 358 (1985), 125-154. MR797679DOI10.1515/crll.1985.358.125
  5. BAOUENDI, M. S. - EBENFELT, P. - ROTHSCHILD, L. P., Real submanifolds in complex space and their mappings, vol. 47, Princeton University Press, Princeton, NJ, 1999. Zbl0944.32040MR1668103DOI10.1515/9781400883967
  6. BLOOM, T. - GRAHAM, I., A geometric characterization of points of type m on real submanifolds of C n , J. Differential Geometry, 12, no. 2 (1977), 171-182. Zbl0436.32013MR492369
  7. BOREL, A. - TITS, J., Groupes réductifs, Inst. Hautes Études Sci. Publ. Math., no. 27 (1965), 55-150. MR207712
  8. BOURBAKI, N., Éléments de mathématique. Fasc. XXXIV. Groupes et algébres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrès par des réflexions. Chapitre VI: Systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968. Zbl0186.33001MR240238
  9. CHARBONNEL, J.-Y. - KHALGUI, H. O., Classification des structures CR invariantes pour les groupes de Lie compacts, J. Lie Theory, 14, no. 1 (2004), 165-198. Zbl1058.22016MR2040175
  10. CHEVALLEY, C., A new kind of relationship between matrices, Amer. J. Math., 65 (1943), 521-531. Zbl0060.03602MR9604DOI10.2307/2371863
  11. CHEVALLEY, C., Algebraic Lie algebras, Ann. of Math. (2), 48 (1947), 91-100. Zbl0032.25202MR19603DOI10.2307/1969217
  12. CHEVALLEY, C., Théorie des groupes de Lie. Tome III. Théorémes généraux sur les algèbres de Lie, Actualités Sci. Ind. no. 1226, Hermann Cie, Paris, 1955. Zbl0186.33104MR68552
  13. FELS, G., Locally homogeneous finitely nondegenerate CR-manifolds, Math. Res. Lett., 14, no. 6 (2007), 893-922. Zbl1155.32027MR2357464DOI10.4310/MRL.2007.v14.n6.a2
  14. GARCÍA, A. N. - SÁNCHEZ, C. U., On extrinsic symmetric CR-structures on the manifolds of complete flags, Beiträge Algebra Geom., 45, no. 2 (2004), 401-414. MR2093174
  15. GILLIGAN, B. - HUCKLEBERRY, A., Fibrations and globalizations of compact homogeneous CR-manifolds, Izv. Math.73 (2009), 501-553. Zbl1172.32011MR2553089DOI10.1070/IM2009v073n03ABEH002455
  16. GOTÔ, M., On algebraic Lie algebras, J. Math. Soc. Japan, 1 (1948), 29-45. MR28306DOI10.2969/jmsj/00110029
  17. HOCHSCHILD, G., An addition to Ado's theorem, Proc. Amer. Math. Soc., 17 (1966), 531-533. Zbl0143.05205MR194482DOI10.2307/2035206
  18. HOCHSCHILD, G. P., Basic theory of algebraic groups and Lie algebras, Graduate Texts in Mathematics, vol. 75, Springer-Verlag, New York, 1981. Zbl0589.20025MR620024
  19. HUMPHREYS, J. E., Introduction to Lie algebras and representation theory, Springer-Verlag, New York, 1972, Graduate Texts in Mathematics, Vol. 9. Zbl0254.17004MR323842
  20. KAUP, W. - ZAITSEV, D., On symmetric Cauchy-Riemann manifolds, Adv. Math., 149, no. 2 (2000), 145-181. Zbl0954.32016MR1742704DOI10.1006/aima.1999.1863
  21. KNAPP, A. W., Lie groups beyond an introduction, second ed., Progress in Mathematics, vol. 140, Birkhäuser Boston Inc. (Boston, MA, 2002). Zbl1075.22501MR1920389
  22. LOEB, J.-J. - MANJARÍN, M. - NICOLAU, M., Complex and CR-structures on compact Lie groups associated to abelian actions, Ann. Global Anal. Geom., 32, no. 4 (2007), 361-378. Zbl1132.32008MR2346223DOI10.1007/s10455-007-9067-7
  23. LOTTA, A. - NACINOVICH, M., On a class of symmetric CR manifolds, Adv. Math., 191, no. 1 (2005), 114-146. Zbl1068.53035MR2102845DOI10.1016/j.aim.2004.03.005
  24. LOTTA, A. - NACINOVICH, M., CR-admissible Z2-gradations and CR-symmetries, Ann. Mat. Pura Appl. (4), 187, no. 2 (2008), 221-236. Zbl1223.32021MR2372800DOI10.1007/s10231-007-0042-5
  25. MEDORI, C. - NACINOVICH, M., Classification of semisimple Levi-Tanaka algebras, Ann. Mat. Pura Appl. (4), 174 (1998), 285-349. Zbl0999.17039MR1746933DOI10.1007/BF01759376
  26. MEDORI, C. - NACINOVICH, M., Complete nondegenerate locally standard CR manifolds, Math. Ann., 317, no. 3 (2000), 509-526. Zbl1037.32031MR1776115DOI10.1007/PL00004412
  27. MEDORI, C. - NACINOVICH, M., The Levi-Malcev theorem for graded CR Lie algebras, Recent advances in Lie theory (Vigo, 2000), Res. Exp. Math., vol. 25 (Heldermann, Lemgo, 2002), 341-346. Zbl1028.32017MR1937989
  28. MEDORI, C. - NACINOVICH, M., Algebras of infinitesimal CR automorphisms, J. Algebra, 287, no. 1 (2005), 234-274. Zbl1132.32013MR2134266DOI10.1016/j.jalgebra.2005.01.030
  29. MOSTOW, G. D., Fully reducible subgroups of algebraic groups, Amer. J. Math., 78 (1956), 200-221. Zbl0073.01603MR92928DOI10.2307/2372490
  30. A. L. ONISHCHIK (ed.), Lie groups and Lie algebras. I, Encyclopaedia of Mathematical Sciences, vol. 20 (Springer-Verlag, Berlin, 1993). 
  31. ROSSI, H., Homogeneous strongly pseudoconvex hypersurfaces, Rice Univ. Studies, 59, no. 1 (1973), 131-145, Complex analysis, 1972 (Proc. Conf., Rice Univ., Houston, Tex., 1972), Vol I: Geometry of singularities. MR330514
  32. SNOW, D. M., Invariant complex structures on reductive Lie groups, J. Reine Angew. Math., 371 (1986), 191-215. Zbl0588.22007MR859325DOI10.1515/crll.1986.371.191
  33. È. B. VINBERG (ed.), Lie groups and Lie algebras, III, Encyclopaedia of Mathematical Sciences, vol. 41 (Springer-Verlag, Berlin, 1994). 
  34. WARNER, G., Harmonic analysis on semi-simple Lie groups I (Springer-Verlag, New York, 1972). Zbl0265.22020MR498999

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.