Proceedings of the Japan Academy, Series A, Mathematical Sciences

Relative versions of theorems of Bogomolov and Sukhanov over perfect fields

Dao Phuong Bac and Nguyen Quoc Thang

Full-text: Open access

Abstract

In this paper, we investigate some aspects of representation theory of reductive groups over non-algebraically closed fields. Namely, we state and prove relative versions of well-known theorems of Bogomolov and of Sukhanov, which are related to observable and quasi-parabolic subgroups of linear algebraic groups over non-algebraically closed perfect fields.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 84, Number 7 (2008), 101-106.

Dates
First available in Project Euclid: 17 July 2008

Permanent link to this document
https://projecteuclid.org/euclid.pja/1216308250

Digital Object Identifier
doi:10.3792/pjaa.84.101

Mathematical Reviews number (MathSciNet)
MR2450060

Zentralblatt MATH identifier
1167.14030

Subjects
Primary: 14L24: Geometric invariant theory [See also 13A50]
Secondary: 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17] 20G15: Linear algebraic groups over arbitrary fields

Keywords
Instability observable subgroups quasi-parabolic subgroups

Citation

Bac, Dao Phuong; Thang, Nguyen Quoc. Relative versions of theorems of Bogomolov and Sukhanov over perfect fields. Proc. Japan Acad. Ser. A Math. Sci. 84 (2008), no. 7, 101--106. doi:10.3792/pjaa.84.101. https://projecteuclid.org/euclid.pja/1216308250


Export citation

References

  • A. Asok, B. Doran and F. Kirwan, Yang-Mills theory and Tamagawa numbers: The fascination of unexpected links in mathematics, Bull. London Math. Soc. 2008.
  • A. Bialinycki-Birula, On homogeneous affine spaces of linear algebraic groups, Amer. J. Math. 85 (1963), 577–582.
  • F. Bien and A. Borel, Sous-groupes épimorphiques de groupes linéaires algébriques. I, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 6, 649–653.
  • A. Borel, Linear algebraic groups, Second edition, Springer, New York, 1991.
  • A. Borel and J. Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. No. 27 (1965), 55–150.
  • A. Borel and J. Tits, Compléments à l'article: “Groupes réductifs”, Inst. Hautes Études Sci. Publ. Math. No. 41 (1972), 253–276.
  • F. A. Bogomolov, Holomorphic tensors and vector bundles on projective varieties, Math. U. S. S. R. Izvestiya 13 (1979), 499–555.
  • F. Châtelet, Variations sur un thème de H. Poincaré, Ann. Sci. École Norm. Sup. (3) 61 (1944), 249–300.
  • F. Coiai and Y. Holla, Extension of structure group of principal bundle in positive characteristic, J. Reine Angew. Math. 595 (2006), 1–24.
  • F. D. Grosshans, Algebraic Homogeneous Spaces and Invariant Theory. Lec. Notes in Math. 1673. Springer, Verlag, 1997.
  • G. R. Kempf, Instability in invariant theory, Ann. of Math. (2) 108 (1978), no. 2, 299–316.
  • S. Mukai, An introduction to invariants and moduli, Translated from the 1998 and 2000 Japanese editions by W. M. Oxbury, Cambridge Univ. Press, Cambridge, 2003.
  • D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory, Third edition, Springer, Berlin, 1994.
  • S. Ramanan and A. Ramanathan, Some remarks on instability flag, Tohoku Math. J. (2) 36 (1984), no. 2, 269–291.
  • C. S. Seshadri, Geometric reductivity over arbitrary base, Advances in Math. 26 (1977), no. 3, 225–274.
  • A. A. Sukhanov, Description of the observable subgroups of linear algebraic groups, Mat. Sb. (N.S.) 137(179) (1988), no. 1, 90–102, 144; translation in Math. USSR-Sb. 65 (1990), no. 1, 97–108.
  • N. Q. Thǎńg and D. P. Bǎć, Some rationality properties of observable groups and related questions, Illinois J. Math. 49 (2005), no. 2, 431–444.
  • J. Tits, Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque, J. Reine Angew. Math. 247 (1971), 196–220.
  • B. Weiss, Finite dimensional representations and subgroup actions on homogeneous spaces, Israel J. Math. 106 (1998), 189–207.