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Mathematical Notes

, Volume 102, Issue 1–2, pp 60–67 | Cite as

Borel subgroups of Cremona groups

  • V. L. Popov
Article

Abstract

We prove that the affine-triangular subgroups are Borel subgroups of Cremona groups.

Keywords

the Cremona group triangular subgroup affine-triangular subgroup the Borel subgroup 

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References

  1. 1.
    J.-P. Serre, “Le groupe de Cremona et ses sous-groupes finis,” Astérisque 332, 75–100 (2010).zbMATHMathSciNetGoogle Scholar
  2. 2.
    J. Blanc, “Groupes de Cremona, connexitéet simplicité,” Ann. Sci. Éc. Norm. Supér. (4) 43 (2), 357–364 (2010).CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    J. Blanc and J.-P. Furter, “Topologies and structures of the Cremona groups,” Ann. of Math. (2) 178 (3), 1173–1198 (2013).CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    C. P. Ramanujam, “A note on automorphism groups of algebraic varieties,” Math. Ann. 156, 25–33 (1964).CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    V. L. Popov, “On infinite dimensional algebraic transformation groups,” Transform. Groups 19 (2), 549–568 (2014).CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    M. Demazure, “Topologies and structures of the Cremona groups,” Ann. Sci. École Norm. Supér. (4) 3, 507–588 (1970).CrossRefzbMATHGoogle Scholar
  7. 7.
    V. L. Popov, “Some subgroups of the Cremona groups,” in Affine Algebraic Geometry (World Sci. Publ., Singapore, 2013), pp. 213–242.CrossRefGoogle Scholar
  8. 8.
    V. L. Popov, “Tori in the Cremona groups,” Izv. Ross. Akad. Nauk Ser. Mat. 77 (4), 103–134 (2013) [Russian Acad. Sci. Izv. Math. 77 (4), 742–771 (2013)].CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    J.-P. Furter and P.-M. Poloni, “On the maximality of the triangular subgroup”, arXiv: 1605.06344 (2016).Google Scholar
  10. 10.
    B. L. van der Waerden, Algebra (Teil I, II, Die Grundlehren der mathematischen Wissenschaften, Bd. 34, Springer-Verlag, Berlin–Göttingen–Heidelberg, 1959, 1960; Vols. 1 and 2, Frederick Ungar Publishing Co., New York, 1970; Nauka, Moscow, 1976).CrossRefzbMATHGoogle Scholar
  11. 11.
    D. A. Suprunenko, Matrix Groups (Nauka, Moscow, 1972; AMS, Providence, RI, 1976).zbMATHGoogle Scholar
  12. 12.
    S. Akbari, R. Ebrahimian, H. Momenaee Kermani, and A. Salehi Golsefidy, “Maximal subgroups of GLn(D),” J. Algebra 259 (1), 201–225 (2003).CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    A. Borel, Linear Algebraic Groups, in Grad. Texts in Math. (Springer-Verlag, New York, 1991), Vol. 126.Google Scholar
  14. 14.
    J. E. Humphreys, Linear Algebraic Groups, in Grad. Texts in Math. (Springer-Verlag, New York, 1975), Vol. 21.Google Scholar
  15. 15.
    T. A. Springer, Linear Algebrac Groups, in Progr. Math. (Birkhäuser Boston, Boston, MA, 1998), Vol. 9.Google Scholar
  16. 16.
    Y. Berest, A. Eshmatov, and F. Eshmatov, “Dixmier groups and Borel subgroups,” Adv. Math. 286, 387–429 (2016).CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    V. L. Popov, “On actions of G a on A n,” in Algebraic Groups, Lecture Notes in Math. (Springer-Verlag, Berlin, 1987), Vol. 1271, pp. 237–242.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

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