Advertisement

Basic reductions in the description of normal subgroups

  • N. A. Vavilov
  • A. K. Stavrova
Article

Abstract

Classification of subgroups in a Chevalley group G(Φ, R) over a commutative ring R, normalized by the elementary subgroup E(Φ, R), is well known. However, for exceptional groups, in the available literature neither the parabolic reduction nor the level reduction can be found. This is due to the fact that the Abe-Suzuki-Vaserstein proof relied on localization and reduction modulo the Jacobson radical. Recently, for the groups of types E 6, E 7, and F 4, the first-named author, M. Gavrilovich, and S. Nikolenko have proposed an even more straightforward geometric approach to the proof of structure theorems, similar to that used for exceptional cases. In the present paper, we give still simpler proofs of two key auxiliary results of the geometric approach. First, we carry through the parabolic reduction in full generality: for all parabolic subgroups of all Chevalley groups of rank ≥ 2. At that we succeeded in avoiding any reference to the structure of internal Chevalley modules, or explicit calculations of the centralizers of unipotent elements. Second, we prove the level reduction, also for the most general situation of double levels, which arise for multiply-laced root systems. Bibliography: 64 titles.

Keywords

Normal Subgroup Commutative Ring General Situation Exceptional Case Simple Proof 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    E. Abe, “Automorphisms of Chevalley groups over commutative rings,” Algebra Analiz, 5, No. 2, 74–90 (1993).Google Scholar
  2. 2.
    H. Bass, Algebraic K-Theory [Russian translation], Mir, Moscow (1973).Google Scholar
  3. 3.
    A. Borel, “Properties and linear representations of Chevalley groups” [Russian translation],” in: Seminar on Algebraic Groups, Mir, Moscow (1973), pp. 9–59.Google Scholar
  4. 4.
    N. Bourbaki, Lie Groups and Algebras [Russian translation], Chaps IV–VI, Moscow (1972).Google Scholar
  5. 5.
    N. A. Vavilov, “Computations in exceptional groups,” Vestn. Samara Univ., 33, No. 7, 11–24 (2007).Google Scholar
  6. 6.
    N. A. Vavilov and M. R. Gavrilovich, “An A2-proof of structure theorems for Chevalley groups of types E6 and E7,” Algebra Analiz, 16, No. 4, 54–87 (2004).MathSciNetGoogle Scholar
  7. 7.
    N. A. Vavilov, M. R. Gavrilovich, and S. I. Nikolenko, “The structure of Chevalley groups: the proof from the Book,” Zap. Nauchn. Semin. POMI, 330, 36–76 (2006).MATHGoogle Scholar
  8. 8.
    N. A. Vavilov and A. Yu. Luzgarev, “The normalizer of the Chevalley group of type E6,” Algebra Analiz, 19, No. 5, 35–62 (2007).MathSciNetGoogle Scholar
  9. 9.
    N. A. Vavilov and S. I. Nikolenko, “An A2-proof of structure theorems for Chevalley groups of type F4,” Algebra Analiz, 20, No. 3 (2008).Google Scholar
  10. 10.
    N. A. Vavilov, E. B. Plotkin, and A. V. Stepanov, “Computations in Chevalley groups over commutative rings,” Dokl. AN SSSR, 40, No. 1, 145–147 (1990).MathSciNetGoogle Scholar
  11. 11.
    N. A. Vavilov and A. V. Stepanov, “The standard commutation formula,” Vestn. S.-Peterburg Gos. Univ., Ser. 1, No. 1, 9–14 (2008).Google Scholar
  12. 12.
    I. Z. Golubchik, “Normal subgroups of linear and unitary groups over an associative ring,” in: Spaces over Algebras and Some Problems in the Theory of Nets, Ufa (1985), pp. 122–142.Google Scholar
  13. 13.
    I. Z. Golubchik, “Lie type groups over PI-rings,” Fund. Prikl. Mat., 3, No. 2, 399–424 (1997).MATHMathSciNetGoogle Scholar
  14. 14.
    V. G. Kazakevich and A. K. Stavrova, “Subgroups normalized by the derived group of the Levi subgroup,” Zap. Nauchn. Semin. POMI, 319, 199–215 (2004).MATHGoogle Scholar
  15. 15.
    R. Steinberg, Lectures on Chevalley Groups [Russian translation], Mir, Moscow (1975).MATHGoogle Scholar
  16. 16.
    A. V. Stepanov, “On the normal structure of the general linear group over the ring of polynomials,” Zap. Nauchn. Semin. POMI, 236, 166–182 (1997).Google Scholar
  17. 17.
    A. A. Suslin, “On the structure of the special linear group over the ring of polynomials,” Izv. AN SSSR, Ser. Mat., 141, No. 2, 235–253 (1977).MathSciNetGoogle Scholar
  18. 18.
    E. Abe, “Chevalley groups over local rings,” Tôhoku Math. J. (2), 21, No. 3, 474–494 (1969).MATHCrossRefGoogle Scholar
  19. 19.
    E. Abe, “Chevalley groups over commutative rings,” in: Proceedings of the Conference on Radical Theory (Sendai, 1988), Uchida Rokakuho, Tokyo (1989), pp. 1–23.Google Scholar
  20. 20.
    E. Abe, “Normal subgroups of Chevalley groups over commutative rings,” Contemp. Math., 83, 1–17 (1989).Google Scholar
  21. 21.
    E. Abe, “Chevalley groups over commutative rings: normal subgroups and automorphisms rings,” Contemp. Math., 184, 13–23 (1995).Google Scholar
  22. 22.
    E. Abe and J. Hurley, “Centers of Chevalley groups over commutative rings,” Comm. Algebra, 16, No. 1, 57–74 (1998).CrossRefMathSciNetGoogle Scholar
  23. 23.
    E. Abe and K. Suzuki, “On normal subgroups of Chevalley groups over commutative rings,” Tôhoku Math. J., 28, No. 1, 185–198 (1976).MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    H. Azad, M. Barry, and G. M. Seitz, “On the structure of parabolic subgroups,” Comm. Algebra, 18, 551–562 (1990).MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    A. Bak, “The stable structure of quadratic modules,” Thesis, Columbia Univ. (1969).Google Scholar
  26. 26.
    A. Bak, “Subgroups of the general linear group normalized by relative elementary groups,” Lecture Notes Math., 967, 1–22 (1982).CrossRefMathSciNetGoogle Scholar
  27. 27.
    A. Bak, “Nonabelian K-theory: The nilpotent class of K1 and general stability,” K-Theory, 4, 363–397 (1991).MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    A. Bak, R. Hazrat, and N. Vavilov, “Localization-completion strikes again: relative K 1 is nilpotent by abelian,” K-Theory (2008).Google Scholar
  29. 29.
    A. Bak and N. Vavilov, “Normality for elementary subgroups functors,” Math. Proc. Cambridge Phil. Soc., 118, No. 1, 35–47 (1995).MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    A. Bak and N. Vavilov, “Structure of hyperbolic unitary groups. I. Elementary subgroups,” Algebra Colloquium, 7, No. 2, 159–196 (2000).MATHMathSciNetGoogle Scholar
  31. 31.
    A. Bak, R. Hazrat, and N. Vavilov, “Structure of hyperbolic unitary groups. II. Normal subgroups” Algebra Colloquium (to appear).Google Scholar
  32. 32.
    H. Bass, “K-theory and stable algebra,” Inst. Hautes Études Sci. Publ. Math., No. 22, 5–60 (1964).Google Scholar
  33. 33.
    H. Bass, “Unitary algebraic K-theory,” Lecture Notes Math., 343, 57–265 (1973).CrossRefMathSciNetGoogle Scholar
  34. 34.
    R. Carter, Simple groups of Lie Type, John Wiley, London et al. (1972).MATHGoogle Scholar
  35. 35.
    D. L. Costa and G. E. Keller, “The E(2, A) sections of SL(2, A),” Ann. Math. (2), 134, No. 1, 159–188 (1991).MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    D. L. Costa and G. E. Keller, “Radix redux: normal subgroups of symplectic groups,” J. Reine Angew. Math., 427, No. 1, 51–105 (1991).MathSciNetGoogle Scholar
  37. 37.
    D. L. Costa and G. E. Keller, “On the normal subgroups of G2(A),” Trans. Amer. Math. Soc., 351, No. 12, 5051–5088 (1999).MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    M. Demazure and A. Grothendieck, “Schémas en groupes. I, II, III,” Lecture Notes Math., 151, 1–564 (1971); 152, 1–654; 153, 1–529.CrossRefGoogle Scholar
  39. 39.
    A. J. Hahn and O. T. O’Meara, The Classical Groups and K-Theory, Springer-Verlag, Berlin et al. (1989).Google Scholar
  40. 40.
    R. Hazrat, “Dimension theory and non-stable K1 of quadratic module,” K-Theory, 27, 293–327 (2002).MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    R. Hazrat and N. Vavilov, “K1 of Chevalley groups are nilpotent,” J. Pure Appl. Algebra, 179, 99–116 (2003).MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    R. Hazrat and N. Vavilov, “Bak’s work on K-theory of rings” (to appear).Google Scholar
  43. 43.
    Li Fuan and Liu Mulan, “Generalized sandwich theorem,” K-Theory, 1, 171–184 (1987).CrossRefMathSciNetGoogle Scholar
  44. 44.
    H. Matsumoto, “Sur les sous-groupes arithmétiques des groupes semi-simples déployés,” Ann. Sci. École Norm. Sup. (4), 2, 1–62 (1969).MATHGoogle Scholar
  45. 45.
    V. M. Petechuk, “Stability structure of linear group over rings,” Mat. Studii, 16, No. 1, 13–24 (2001).MATHMathSciNetGoogle Scholar
  46. 46.
    G. E. Roehrle, “On the structure of parabolic subgroups in algebraic groups,” J. Algebra, 157, No. 1, 80–115 (1993).MATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    A. Stavrova, “Normal structure of maximal parabolic subgroups in Chevalley groups over rings,” Algebra Colloquium (to appear).Google Scholar
  48. 48.
    M. R. Stein, “Generators, relations, and coverings of Chevalley groups over commutative rings,” Amer. J. Math., 93, No. 4, 965–1004 (1971).MATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    A. Stepanov and N. Vavilov, “Decomposition of transvections: a theme with variations,” K-Theory, 19, 109–153 (2000).MATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    K. Suzuki, “On normal subgroups of twisted Chevalley groups over local rings,” Sci. Rep. Tokyo Kyoiku Daigaku, 13, 237–249 (1997).Google Scholar
  51. 51.
    K. Suzuki, “Normality of the elementary subgroups of twisted Chevalley groups over commutative rings,” J. Algebra, 175, No. 3, 526–536 (1995).MATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    G. Taddei, “Normalité des groupes élémentaire dans les groupes de Chevalley sur un anneau,” Contemp. Math., 55, 693–710 (1986).MathSciNetGoogle Scholar
  53. 53.
    L. N. Vaserstein, “On the normal subgroups of the GLn of a ring,” Lecture Notes Math., 854, 454–465 (1981).MathSciNetGoogle Scholar
  54. 54.
    L. N. Vaserstein, “The subnormal structure of general linear groups,” Math. Proc. Cambridge Phil. Soc., 99, 425–431 (1986).MATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    L. N. Vaserstein, “On normal subgroups of Chevalley groups over commutative rings,” Tôhoku Math. J. (2), 36, No. 5, 219–230 (1986).CrossRefMathSciNetGoogle Scholar
  56. 56.
    L. N. Vaserstein, “The subnormal structure of general linear groups over rings,” Math. Proc. Cambridge Phil. Soc., 108, No. 2, 219–229 (1990).MATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    L. N. Vaserstein and You Hong, “Normal subgroups of classical groups over rings,” J. Pure Appl. Algebra, 105, No. 1, 93–106 (1995).MATHCrossRefMathSciNetGoogle Scholar
  58. 58.
    N. Vavilov, “A note on the subnormal structure of general linear groups,” Math. Proc. Cambridge Phil. Soc., 107, No. 2, 193–196 (1990).MATHCrossRefMathSciNetGoogle Scholar
  59. 59.
    N. Vavilov, “Structure of Chevalley groups over commutative rings,” in: Proceedings of the Conference on nonassociative Algebras and Related Topics (Hiroshima, 1990), World Sci. Publ., London et al. (1991), pp. 219–335.Google Scholar
  60. 60.
    N. Vavilov, “A third look at weight diagrams,” Rendiconti del Rend. Sem. Mat. Univ. Padova, 204, No. 1, 201–250 (2000).MathSciNetGoogle Scholar
  61. 61.
    N. Vavilov, “An A3-proof of structure theorems for Chevalley groups of types E6 and E7,” Intern. J. Algebra Comput., 17, No. 5–6, 1283–11298 (2007).MATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    N. Vavilov, “Structure of exceptional groups over rings,” in: Proceedings of the 3rd International Congress on Algebra and Combinatorics (Beijing and Xian, 2007) (to appear).Google Scholar
  63. 63.
    N. A. Vavilov and E. B. Plotkin, “Chevalley groups over commutative rings. I. Elementary calculations,” Acta Appl. Math., 45, 73–115 (1996).MATHCrossRefMathSciNetGoogle Scholar
  64. 64.
    Zhang, Zuhong, “Stable sandwich classification theorem for classical-like groups,” Math. Proc. Cambridge Phil. Soc., 143, No. 3, 607–619 (2007).MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  • N. A. Vavilov
    • 1
  • A. K. Stavrova
    • 1
  1. 1.St.Petersburg State UniversitySt.PetersburgRussia

Personalised recommendations