Advertisement

Journal of Mathematical Sciences

, Volume 179, Issue 6, pp 662–678 | Cite as

The yoga of commutators

  • R. Hazrat
  • A. V. Stepanov
  • N. A. Vavilov
  • Z. Zhang
Article

In the present paper, we discuss some recent versions of localization methods for calculations in the groups of points of algebraic-like and classical-like groups. Namely, we describe relative localization, universal localization, and enhanced versions of localization-completion. Apart from the general strategic description of these methods, we state some typical technical results of conjugation calculus and commutator calculus. Also, we state several recent results obtained therewith, such as relative standard commutator formulas, bounded width of commutators with respect to the elementary generators, and nilpotent filtrations of congruence subgroups. Overall, this shows that localization methods can be much more efficient than expected. Bibliography: 74 titles.

Keywords

Relative Commutator Algebraic Group Relative Localization Localization Method Elementary Generator 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. Abe, “Whitehead groups of Chevalley groups over polynomial rings,” Comm. Algebra, 11, No. 12, 1271–1308 (1983).CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    E. Abe, “Chevalley groups over commutative rings,” in: Proceedings of the Conference on Radical Theory, Sendai (1988), pp. 1–23.Google Scholar
  3. 3.
    E. Abe, “Normal subgroups of Chevalley groups over commutative rings,” Contemp. Math., 83, 1–17 (1989).Google Scholar
  4. 4.
    A. Bak, “Subgroups of the general linear group normalized by relative elementary groups,” Lect. Notes Math., 967, 1–22 (1982).CrossRefMathSciNetGoogle Scholar
  5. 5.
    A. Bak, “Non-Abelian K-theory: The nilpotent class of K 1 and general stability,” K-theory, 4, 363–397 (1991).CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    A. Bak, R. Basu, and R. A. Rao, “Local–global principle for transvection groups,” Proc. Amer. Math. Soc., 138, No. 4, 1191–1204 (2010),CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    A. Bak, R. Hazrat, and N. Vavilov, “Localization-completion strikes again: relative K 1 is nilpotent by Abelian,” J. Pure Appl. Algebra, 213, 1075–1085 (2009).CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    A. Bak and N. Vavilov, “Normality for elementary subgroup functors,” Math. Proc. Cambridge Phil. Soc., 118, No. 1, 35–47 (1995).CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    A. Bak and N. Vavilov, “Structure of hyperbolic unitary groups. I. Elementary subgroups,” Algebra Colloq., 7, No. 2, 159–196 (2000).CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    R. Basu, R. A. Rao, and R. Khanna, “On Quillen’s local global principle,” Contemp. Math., 390, 17–30 (2005).MathSciNetGoogle Scholar
  11. 11.
    R. K. Dennis and L. N. Vaserstein, “On a question of M. Newman on the number of commutators,” J. Algebra, 118, 150–161 (1988).CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    R. K. Dennis and L. N. Vaserstein, “Commutators in linear groups.” K-Theory, 2, 761–767 (1989).CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    E. Ellers and N. Gordeev, “On the conjectures of J. Thompson and O. Ore,” Trans. Amer. Math. Soc., 350, 3657–3671 (1998).CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    G. Habdank, “A classification of subgroups of Λ-quadratic groups normalized by relative elementary sub-groups,” Dissertation, Universität Bielefeld (1987).Google Scholar
  15. 15.
    G. Habdank, “A classification of subgroups of Λ-quadratic groups normalized by relative elementary sub-groups,” Adv. Math., 110, No. 2, 191–233 (1995). CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    A. J. Hahn and O. T. O’Meara, The Classical Groups and K-Theory, Springer-Verlag, Berlin (1989).MATHGoogle Scholar
  17. 17.
    R. Hazrat, “Dimension theory and nonstable K 1 of quadratic module,” K-theory, 27, 293–327 (2002).CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    R. Hazrat, “On K-theory of classical-like groups,” Doktorarbeit, Universität, Bielefeld, 1–62 (2002).Google Scholar
  19. 19.
    R. Hazrat, V. Petrov, and N. Vavilov, “Relative subgroups in Chevalley groups,” J. K-theory, 5, 603–618 (2010).CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    R. Hazrat, A. Stepanov, N. Vavilov, and Z. Zhang, “New versions of localization method,” in: Topology, Geometry, and Dynamics. Rokhlin Memorial, St.Petersburg (2010), pp. 114–116.Google Scholar
  21. 21.
    R. Hazrat, A. Stepanov, N. Vavilov, and Z. Zhang, “On the length of commutators in unitary groups,” Preprint (2010).Google Scholar
  22. 22.
    R. Hazrat and N. Vavilov, “K l of Chevalley groups arc nilpotent,” J. Pure Appl. Algebra, 179, 99–116 (2003).CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    R. Hazrat and N. Vavilov, “Bak’s work on K-theory of rings (with an appendix by Max Karoubi),” J. K-Theory, 4, No. 1 (2009), 1–65.CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    R. Hazrat, N. Vavilov, and Z. Zhang, “Relative commutator calculus in unitary groups, and applications,” J. Algebra, 343, No. 1, 107–137 (2010).CrossRefGoogle Scholar
  25. 25.
    R. Hazrat, N. Vavilov, and Z. Zhang, “Relative commutator calculus in Chevalley groups, and applications,” arXiv:1107/3009v1, submitted to J. Algebra.
  26. 26.
    R. Hazrat and Z. Zhang, “Generalized commutator formula,” Comm. Algebra, 39, 1441–1454 (2011).CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    W. van der Kallen, “SL3(\( \mathbb{C} \)[x]) does not have bounded word length,” Lect. Notes Math., 966, 357–361 (1982).CrossRefGoogle Scholar
  28. 28.
    M.-A. Knus, Quadratic and Hermitian Forms Over Rings, Springer-Verlag, Berlin (1991).MATHGoogle Scholar
  29. 29.
    V. I. Kopeiko, “The stabilization of symplectic groups over a polynomial ring,” Math. USSR Sbornik, 34, 655–669 (1978).CrossRefGoogle Scholar
  30. 30.
    T. Y. Lam, Serre’s Problem on Projective Modules, Springer-Verlag, Berlin (2006).CrossRefGoogle Scholar
  31. 31.
    L. Fuan, “The structure of symplectic group over arbitrary commutative rings,” Acta Math. Sinica, 3, No. 3, 247–255 (1987).CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    L. Fuan, “The structure of orthogonal groups over arbitrary commutative rings,” Chinese Ann. Math. Ser. B, 10, No. 3, 341–350 (1989).MATHMathSciNetGoogle Scholar
  33. 33.
    L. Fuan and L. Mulan, “Generalized sandwich theorem,” K-Theory, 1 (1987), 171–184.CrossRefMathSciNetGoogle Scholar
  34. 34.
    M. Liebeck, E. A. O’Brien, A. Shalev, and Pham Huu Tiep, “The Ore conjecture,” Preprint (2009); http://www.math.auckland.ac.nz/obrien/research/ore.pdf.
  35. 35.
    A. Yu. Luzgarev, “Overgroups of E(F 4, R) in G(E 6, R),” St.Petersburg J. Math., 20, No. 5, 148–185 (2008).Google Scholar
  36. 36.
    A. Yu. Luzgarev and A. K. Stavrova, “Elementary subgroups of isotropic reductive groups are perfect,” Algebra Analiz, 23, No. 5, 140–154 (2011).Google Scholar
  37. 37.
    A. W. Mason, “A note on subgroups of GL(n, A) which are generated by commutators,” J. London Math. Soc., 11, 509–512 (1974).CrossRefGoogle Scholar
  38. 38.
    A. W. Mason, “On subgroups of GL(n, A) which are generated by commutators. II,” J. reine angew. Math., 322, 118–135 (1981).CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    A. W. Mason, “A further note on subgroups of GL(n, A) which are generated by commutators,” Arch. Math., 37, No. 5, 401–405 (1981).CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    A. W. Mason and W. W. Stothers, “On subgroups of GL(n, A) which are generated by commutators,” Invent. Math., 23, 327–346 (1974).CrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    D. W. Morris, “Bounded generation of SL(n, A) (after D. Carter, G. Keller, and E. Paige),” New York J. Math., 13, 383–421 (2007).MATHMathSciNetGoogle Scholar
  42. 42.
    V. Petrov, “Overgroups of unitary groups,” K-Theory, 29, 147–174 (2003).CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    V. A. Petrov, “Odd unitary groups,” J. Math. Sci., 130, No. 3, 4752–4766 (2003).CrossRefGoogle Scholar
  44. 44.
    V. A. Petrov, “Overgroups of classical groups,” Ph.D. Thesis, St.Petersburg State Univeresity (2005).Google Scholar
  45. 45.
    V. A. Petrov and A. K. Stavrova, “Elementary subgroups of isotropic reductive groups,” St.Petersburg Math. J., 20, No. 3, 160–188 (2008).MathSciNetGoogle Scholar
  46. 46.
    A. Sivatski and A. Stepanov, “‘On the word length of commutators in GLn(R),” K-theory, 17, 295–302 (1999).CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    A. Stepanov, “Universal localization in algebraic groups,” http://alexei.stepanov.spb.ru/publicat.html (2010).
  48. 48.
    A. Stepanov and N. Vavilov, “Decomposition of transvections: a theme with variations,” K-Theory, 19 (2000), 109–153.CrossRefMATHMathSciNetGoogle Scholar
  49. 49.
    A. Stepanov and N. Vavilov, “On the length of commutators in Chevalley groups,” Israel J. Math., 185, No. 1, 253–276 (2011).CrossRefGoogle Scholar
  50. 50.
    A. Stepanov, N. Vavilov, and Y. Hong, “Overgroups of semi-simple subgroups via localisation-completion,” Preprint (2011).Google Scholar
  51. 51.
    A. A. Suslin, “The structure of the special linear group over polynomial rings,” Math. USSR Izv., 11, No. 2, 235–253 (1977).CrossRefMathSciNetGoogle Scholar
  52. 52.
    A. A. Suslin and V. I. Kopeiko, “Quadratic modules and orthogonal groups over polynomial rings,” J. Sov. Math., 20, No. 6, 2665–2691 (1982).CrossRefMATHGoogle Scholar
  53. 53.
    G. Taddei, “Schémas de Chevalley-Demazure, fonctions représentatives et théorème de normalité,” Thèse, Université de Genève (1985).Google Scholar
  54. 54.
    G. Taddei, “Normalité des groupes élémentaires dans les groupes de Chevalley sur un anneau,” Contemp. Math., 55, No. 2, 693–710 (1986).MathSciNetGoogle Scholar
  55. 55.
    M. S. Tulenbaev, “The Steinberg group of a polynomial ring,” Math. USSR Sb., 45, No. 1, 139–154 (1983).CrossRefMATHGoogle Scholar
  56. 56.
    L. Vaserstein, “On the normal subgroups of the GLn of a ring,” in: Algebraic K- Theory (Evanston, 1980), Lect. Notes Math., 854, 454–465 (1981).Google Scholar
  57. 57.
    L. Vaserstein, “The subnormal structure of general linear groups.” Math. Proc. Cambridge Phil. Soc., 99, 425–431 (1986).CrossRefMATHMathSciNetGoogle Scholar
  58. 58.
    L. Vaserstein, “On normal subgroups of Chevalley groups over commutative rings,” Tôhoku Math. J., 36, No. 5, 219–230 (1986).CrossRefMathSciNetGoogle Scholar
  59. 59.
    L. Vaserstein, “Normal subgroups of orthogonal groups over commutative rings,” Amer. J. Math., 110, No. 5, 955–973 (1988).CrossRefMATHMathSciNetGoogle Scholar
  60. 60.
    L. Vaserstein, “Normal subgroups of symplectic groups over rings,” K-Theory, 2, No. 5, 647–673 (1989).CrossRefMATHMathSciNetGoogle Scholar
  61. 61.
    L. Vaserstein, “The subnormal structure of general linear groups over rings,” Math. Proc. Cambridge Phil. Soc., 108, No. 2, 219–229 (1990).CrossRefMATHMathSciNetGoogle Scholar
  62. 62.
    L. Vaserstein and Y. Hong, “Normal subgroups of classical groups over rings,” J. Pure Appl. Algebra, 105, No. 1, 93–106 (1995).CrossRefMATHMathSciNetGoogle Scholar
  63. 63.
    N. Vavilov, “A note on the subnormal structure of general linear groups,” Math. Proc. Cambridge Phil. Soc., 107, No. 2, 193–196 (1990).CrossRefMATHMathSciNetGoogle Scholar
  64. 64.
    N. Vavilov, “Structure of Chevalley groups over commutative rings,” in: Nonassociative Algebras and Related Topics (Hiroshima, 1990), World Sci. Publ., River Edge, New Jersey (1991), pp. 219–335.Google Scholar
  65. 65.
    N. Vavilov, A. Luzgarev, and A. Stepanov, “Calculations in exceptional groups over rings.” J. Math. Sci., 373, 48–72 (2009).MathSciNetGoogle Scholar
  66. 66.
    N. A. Vavilov and V. A. Petrov, “Overgroups of Ep(n, R),” St.Petersburg Math. J., 15, No. 4, 515–543 (2004).CrossRefMathSciNetGoogle Scholar
  67. 67.
    N. A. Vavilov and A. V. Stepanov, “Standard commutator formula,” Vestnik St.Petersburg Univ. Math., 41, No. 1, 5–8 (2008).CrossRefMATHMathSciNetGoogle Scholar
  68. 68.
    N. A. Vavilov and A. V. Stepanov, “Overgroups of semi-simple groups,” Vestnik Samara State Univ., Ser. Nat. Sci., No. 3, 51–95 (2008).Google Scholar
  69. 69.
    N. A. Vavilov and A. V. Stepanov, “Standard commutator formulae, revisited,” Vestnik St. Petersburq Univ. Math.. 43, No. 1, 12-17 (2010).CrossRefMathSciNetGoogle Scholar
  70. 70.
    N. A. Vavilov and A. V. Stepanov, “Linear groups over general rings. I. Main structure theorems,” Vestnik Samara State Univ., Ser. Nat. Sci., 1–87 (2010).Google Scholar
  71. 71.
    H. You, “Subgroups of classical groups normalised by relative elementary groups,” J. Pure Appl. Algebra, 1–16 (2010).Google Scholar
  72. 72.
    Z. Zhang, “Lower K-theory of unitary groups,” Ph.D. Thesis, University of Belfast (2007).Google Scholar
  73. 73.
    Z. Zhang, “Stable sandwich classification theorem for classical-like groups,” Math. Prac. Cambridge Phil. Soc., 143, No. 3, 607–619 (2007).MATHGoogle Scholar
  74. 74.
    Z. Zhang, “Subnormal structure of nonstable unitary groups over rings,” J. Pure Appl. Algebra, 214, 622–628 (2010).CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • R. Hazrat
    • 1
  • A. V. Stepanov
    • 2
  • N. A. Vavilov
    • 3
  • Z. Zhang
    • 4
  1. 1.Queen’s University BelfastBelfastUnited Kingdom
  2. 2.St.Petersburg State Electrotechnical UniversitySt. PetersburgRussia
  3. 3.St.Petersburg State UniversitySt. PetersburgRussia
  4. 4.Beijing Institute of TechnologyBeijingChina

Personalised recommendations