Acta Applicandae Mathematica

, Volume 45, Issue 1, pp 73–113

Chevalley groups over commutative rings: I. Elementary calculations

  • Nikolai Vavilov
  • Eugene Plotkin


This is the first in a series of papers dedicated to the structure of Chevalley groups over commutative rings. The goal of this series is to systematically develop methods of calculations in Chevalley groups over rings, based on the use of their minimal modules. As an application, we give new direct proofs for normality of the elementary subgroup, description of normal subgroups and similar results due to E. Abe, G. Taddei, L. N. Vaserstein, and others, as well as some generalizations. In this first part we outline the whole project, reproduce construction of Chevalley groups and their elementary subgroups, recall familiar facts about the elementary calculations in these groups, and fix a specific choice of the structure constants.

Mathematics Subject Classifications (1991)

20G35 20G15 

Key words

Chevalley groups over rings elementary subgroup K1-functor Weyl modules Chevalley commutator formula Steinberg relations structure constants 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abe, E.: Chevalley groups over local rings, Tôhoku Math. J. 21(3) (1969), 474–494.Google Scholar
  2. 2.
    Abe, E.: Hopf Algebras, Cambridge Univ. Press, 1980.Google Scholar
  3. 3.
    Abe, E.: Whitehead groups of Chevalley groups over polynomial rings, Comm. Algebra 11(12) (1983), 1271–1308.Google Scholar
  4. 4.
    Abe, E.: Whitehead groups of Chevalley groups over Laurent polynomial rings, Preprint Univ. Tsukuba, 1988.Google Scholar
  5. 5.
    Abe, E.: Chevalley groups over commutative rings, Proc. Conf. Radical Theory (Sendai, 1988), Uchida Rokakuho Publ. Comp., Tokyo, 1989, pp. 1–23.Google Scholar
  6. 6.
    Abe, E.: Normal subgroups of Chevalley groups over commutative rings, Contemp. Math. 83 (1989), 1–17.Google Scholar
  7. 7.
    Abe, E.: Automorphisms of Chevalley groups over commutative rings, Algebra Anal. 5(2) (1993), 74–90 (in Russian).Google Scholar
  8. 8.
    Abe, E. and Hurley, J.: Centers of Chevalley groups over commutative rings, Comm. Algebra 16(1) (1988), 57–74.Google Scholar
  9. 9.
    Abe, E. and Suzuki, K.: On normal subgroups of Chevalley groups over commutative rings, Tôhoku Math. J. 28(1) (1976), 185–198.Google Scholar
  10. 10.
    Artin, E.: Geometric Algebra, Wiley, New York, 1957.Google Scholar
  11. 11.
    Aschbacher, M.: The 27-dimensional module for E6. I, Invent Math. 89(1) (1987), 159–195.Google Scholar
  12. 12.
    Aschbacher, M.: Chevalley groups of type G2 as the groups of a trilinear form, J. Algebra 109(1) (1987), 193–259.Google Scholar
  13. 13.
    Aschbacher, M.: Some multilinear forms with large isometry groups, Geom. Dedicata 25 (1988), 417–465.Google Scholar
  14. 14.
    Azad, H.: Structure constants of algebraic groups, J. Algebra 75(1) (1982), 209–222.Google Scholar
  15. 15.
    Azad, H.: The Jacobi identity, Punjab Univ. J. Math., 16 (1983), 9–29.Google Scholar
  16. 16.
    Bak, A.: The stable structure of quadratic modules, Thesis, Columbia Univ., New York, 1969.Google Scholar
  17. 17.
    Bak, A.: On modules with quadratic forms, in Lecture Notes in Math. 108, Springer, New York, 1969, pp. 55–66.Google Scholar
  18. 18.
    Bak, A.: K-theory of Forms, Princeton Univ. Press, Princeton, New York, 1981.Google Scholar
  19. 19.
    Bak, A.: Nonabelian K-theory: The nilpotent class of K1 and general stability, K-theory 4 (1991), 363–397.Google Scholar
  20. 20.
    Bak, A. and Vavilov, N. A.: Normality for Elementary Subgroup Functors, Cambridge Univ. Press, 1995 (to appear).Google Scholar
  21. 21.
    Bak, A. and Vavilov, N. A.: Structure of hyperbolic unitary groups. I. Elementary subgroups (to appear).Google Scholar
  22. 22.
    Bak, A. and Vavilov, N. A.: Structure of hyperbolic unitary groups. II. Normal subgroups (to appear).Google Scholar
  23. 23.
    Bass, H.: K-theory and stable algebra, in Publ. Math. Inst. Hautes Et. Sci. (1964), No. 22, pp. 5–60.Google Scholar
  24. 24.
    Bass, H.: Algebraic K-theory, Benjamin, New York, 1968.Google Scholar
  25. 25.
    Bass, H.: Unitary algebraic K-theory, in Lecture Notes in Math. 343, Springer, New York, 1973, pp. 57–265.Google Scholar
  26. 26.
    Bass, H.: Clifford algebras and spinor norms over a commutative ring, Amer. J. Math. 96(1) (1974), 156–206.Google Scholar
  27. 27.
    Bass, H., Milnor, J., and Serre, J.-P.: Solution of the congruence subgroup problem for SLn(n≥3) and Sp2n(n≥2), in Publ. Math. Inst. Hautes Et.Sci. (1967), No. 33, pp. 59–137.Google Scholar
  28. 28.
    Bayer-Fluckiger, E.: Principle de Hasse faible pour les systèmes de formes quadratiques, J. Reine Angew. Math. 378(1) (1987), 53–59.Google Scholar
  29. 29.
    Borel, A.: Linear Algebraic Groups, 2nd edn, Springer, New York, 1991.Google Scholar
  30. 30.
    Borel, A.: Properties and linear representations of Chevalley groups, in Lecture Notes in Math. 131, Springer, New York, 1970, pp. 1–55.Google Scholar
  31. 31.
    Borewicz, Z. I. and Vavilov, N. A.: The distribution of subgroups in the general linear group over a commutative ring, in Proc. Steklov Inst. Math. (1985), No. 3, pp. 27–46.Google Scholar
  32. 32.
    Bourbaki, N.: Groupes et algèbres de Lie, Ch. 4–6, Hermann, Paris, 1968.Google Scholar
  33. 33.
    Bourbaki, N.: Groupes et algèbres de Lie, Ch. 7–8, Hermann, Paris, 1975.Google Scholar
  34. 34.
    Brown, R. B.: Groups of type E7, J. Reine Angew. Math., 236(1) (1969), 79–102.Google Scholar
  35. 35.
    Brown, R. B.: A characterization of spin representations, Canad. J. Math. 23(5) (1971), 896–906.Google Scholar
  36. 36.
    Burgoyne, N. and Williamson, C.: Some computations involving simple Lie algebras, in Proc. 2nd Symp. Symbolic and Algebraic Manipulation, Ass. Comp. Mach., New York, 1971.Google Scholar
  37. 37.
    Bürgstein, H. and Hesselink, W. H.: Algorithmic orbit classification for some Borel group actions, Compositio Math. 61(1) (1988), 3–41.Google Scholar
  38. 38.
    Carter, R. W.: Simple groups and simple Lie algebras, J. London Math. Soc. 40(1) (1965), 163–240.Google Scholar
  39. 39.
    Carter, R. W.: Simple Groups of Lie Type, Wiley, London, 1972.Google Scholar
  40. 40.
    Chevalley, C.: Sur le groupe exceptionnel (E6), C.R. Acad. Sci. Paris 232 (1951), 1991–1993.Google Scholar
  41. 41.
    Chevalley, C.: Sur certains groupes simples, Tôhoku Math. J. 7(1) (1955), 14–66.Google Scholar
  42. 42.
    Chevalley, C.: Classification des groupes de Lie algebriques, vol. 2, Secretariat Mathématique, Paris, 1956–1958.Google Scholar
  43. 43.
    Chevalley, C.: Certain schemas des groupes semi-simples, Sem. Bourbaki (1960–1961), No. 219, 1–16.Google Scholar
  44. 44.
    Cohen, A. M. and Cooperstein, B. N.: The 2-spaces of the standard E6(q)-module, Geom. Dedicata 25 (1988), 467–480.Google Scholar
  45. 45.
    Cohen, A. M. and Cushman, R. H.: Gröbner bases and standard monomial theory, in Computational Algebraic Geometry, Progress in Mathematics 109, Birkhäuser, Basel, 1993, pp. 41–60.Google Scholar
  46. 46.
    Cohen, A. M., Griess, R. L., and Lisser, B.: The group L(2, 61) embeds in the Lie group of type E8 (to appear).Google Scholar
  47. 47.
    Cohn, P.: On the structure of the GL2 of a ring, Publ. Math. Inst. Hautes Et. Sci. (1966), No. 30, pp. 365–413.Google Scholar
  48. 48.
    Cooperstein, B. N.: The geometry of root subgroups in exceptional groups, Geom. Dedicata 8 (1978), 317–381; 15 (1983), 1–45.Google Scholar
  49. 49.
    Costa, D. L.: Zero-dimensionality and the GE2 of polynomial rings, J. Pure Appl. Algebra 50 (1988), 223–229.Google Scholar
  50. 50.
    Demazure, M.: Schémes en groupes réductifs, Bull. Soc. Math. France 93 (1965), 369–413.Google Scholar
  51. 51.
    Demazure, M. and Gabriel, P.: Groupes algèbriques, I, North-Holland, Amsterdam, 1970.Google Scholar
  52. 52.
    Demazure, M. and Gabriel, P.: Introduction to Algebraic Geometry and Algebraic Groups, North-Holland, Amsterdam, 1980.Google Scholar
  53. 53.
    Demazure, M. and Grothendieck, A.: Schémes en groupes, I, II, III, Lecture Notes in Math. 151–153, Springer, New York, 1971.Google Scholar
  54. 54.
    Dieudonné, J.: On simple groups of type Bn, Amer. J. Math. 79(5) (1957), 922–923.Google Scholar
  55. 55.
    Dieudonné, J.: La géométrie des groupes classiques, 3rd edn, Springer, Berlin, 1971.Google Scholar
  56. 56.
    Estes, D. and Ohm, J.: Stable range in commutative rings, J. Algebra 7(3) (1967), 343–362.Google Scholar
  57. 57.
    Frenkel, I. B. and Kac, V.: Basic representations of affine Lie algebras and dual resonance models, Invent. Math. 62(1) (1980), 23–66.Google Scholar
  58. 58.
    Frenkel, I. B., Lepowsky, J., and Meurman, A.: Vertex Operator Algebras and the Monster, Academic Press, New York, 1988.Google Scholar
  59. 59.
    Gabriel, P. and Roiter, A. V.: Representations of Finite-Dimensional Algebras, Springer, New York, 1993.Google Scholar
  60. 60.
    Gilkey, P. and Seitz, G.: Some representations of exceptional Lie algebras, Geom. Dedicata 25 (1988), 407–416.Google Scholar
  61. 61.
    Golubchik, I. Z.: On the general linear group over an associative ring, Uspekhi Mat. Nauk 28(3) (1973), 179–180 (in Russian).Google Scholar
  62. 62.
    Golubchik, I. Z.: On the normal subgroups of orthogonal group over an associative ring with involution, Uspekhi Mat. Nauk 30(6) (1975), 165 (in Russian).Google Scholar
  63. 63.
    Golubchik, I. Z.: The normal subgroups of linear and unitary groups over rings, PhD Thesis, Moscow State Univ., 1981 (in Russian).Google Scholar
  64. 64.
    Golubchik, I. Z.: On the normal subgroups of the linear and unitary groups over associative rings, in Spaces over Algebras and Some Problems in the Theory of Nets, Ufa, 1985, pp. 122–142 (in Russian).Google Scholar
  65. 65.
    Golubchik, I. Z. and Mikhalev, A. V.: Isomorphisms of general linear groups over associative rings, Vestnik Moskov, Univ. Ser. I, Mat. Mekh. 38(3) (1983), 73–85.Google Scholar
  66. 66.
    Golubchik, I. Z. and Mikhalev, A. V.: Elementary subgroup of a unitary group over a PI-ring, Vestnik Moskov. Univ. Ser. I, Mat. Mekh. 1 (1985), 30–36.Google Scholar
  67. 67.
    Goto, M. and Grosshans, F. D.: Semi-Simple Lie Algebras, Marcel Dekker, New York, 1978.Google Scholar
  68. 68.
    Grünewald, F., Mennicke, J., and Vaserstein, L. N.: On symplectic groups over polynomial rings, Math. Z. 206(1) (1991), 35–56.Google Scholar
  69. 69.
    Hahn, A. J. and O'Meara, O. T.: The Classical Groups and K-Theory, Springer, Berlin, 1989.Google Scholar
  70. 70.
    Haboush, W. J.: Central differential operators on split semi-simple groups of positive characteristic, in Lecture Notes in Math. 795, Springer, New York, 1980, pp. 35–86.Google Scholar
  71. 71.
    Hée, J.-Y.: Groupes de Chevalley et groupes classiques, Publ Math. Univ. Paris VII 17 (1984), 1–54.Google Scholar
  72. 72.
    Hesselink, W.: A classification of the nilpotent triangular matrices, Compositio Math. 55(1) (1985), 89–133.Google Scholar
  73. 73.
    Humphreys, J. E.: Linear Algebraic Groups, Springer, New York, 1975.Google Scholar
  74. 74.
    Humphreys, J. E.: Introduction to Lie Algebras and Representation Theory, Springer, Berlin, 1980.Google Scholar
  75. 75.
    Humphreys, J. E.: On the hyperalgebra of a semi-simple algebraic group, in Contributions to Algebra, Academic Press, New York, 1977, pp. 203–210.Google Scholar
  76. 76.
    Hurley, J. F.: Some normal subgroups of elementary subgroups of Chevalley groups over rings, Amer. J. Math. 93(4) (1971), 1059–1069.Google Scholar
  77. 77.
    Jacobson, N.: Lie Algebras, Interscience Press, New York, 1962.Google Scholar
  78. 78.
    Jantzen, J. C.: Representations of Algebraic Groups, Academic Press, New York, 1987.Google Scholar
  79. 79.
    Kac, V.: Infinite Dimensional Lie Algebras, 2nd edn, Cambridge Univ. Press, Cambridge, 1985.Google Scholar
  80. 80.
    van der, Kallen, W.: Another presentation for the Steinberg group, Proc. Nederl. Akad. Wetensch., Ser. A 80 (1977), 304–312.Google Scholar
  81. 81.
    Kopeiko, V. I.: The stabilization of symplectic groups over a polynomial ring, Math. USSR Sbornik 34 (1978), 655–669.Google Scholar
  82. 82.
    Kopeiko, V. I.: Unitary and orthogonal groups over rings with involution, in Algebra and Discrete Math., Kalmytsk. Gos. Univ., Elista, 1985, pp. 3–14 (in Russian).Google Scholar
  83. 83.
    Kostant, B.: Groups over ℤ, Proc. Symp. Pure Math. 9 (1966), 90–98.Google Scholar
  84. 84.
    Lakshmibai, V. and Seshadri, C. S.: Geometry of G/P. V, J. Algebra 100 (1986), 462–557.Google Scholar
  85. 85.
    Matsumoto, H.: Sur les sous-groupes arithmétiques des groupes semi-simples deployés, Ann. Sci. Ecole. Norm. Sup. 4 Sér. 2 (1969), 1–62.Google Scholar
  86. 86.
    Milnor, J.: Introduction to Algebraic K-Theory, Princeton Univ. Press, Princeton, New York, 1971.Google Scholar
  87. 87.
    Mizuno, K.: The conjugate classes of Chevalley groups of type E6, J. Fac. Sci. Univ. Tokyo 24(3) (1977), 525–563.Google Scholar
  88. 88.
    Mizuno, K.: The conjugate classes of unipotent elements of the Chevalley groups E7 and E8, Tokyo J. Math. 3(2) (1980), 391–458.Google Scholar
  89. 89.
    Michel, L., Patera, J., and Sharp, R.: The Demazure-Tits subgroup of a simple Lie group, J. Math. Phys. 29(4) (1988), 777–796.Google Scholar
  90. 90.
    Petechuk, V. M.: Automorphisms of matrix groups over commutative rings, Math. USSR Sbornik 45 (1983), 527–542.Google Scholar
  91. 91.
    Plotkin, E. B.: Surjective stability theorems for the K1-functors for some exceptional Chevalley groups, J. Soviet Math. 198 (1993), 751–766.Google Scholar
  92. 92.
    Plotkin, E. B.: Stability theorems for K1-functors for Chevalley groups, in Proc. Conf. Nonassociative Algebras and Related Topics, Hiroshima, 1990, World Scientific, Singapore, 1991, pp. 203–217.Google Scholar
  93. 93.
    Plotkin, E. B. and Vavilov, N. A.: Stability theorems for K1- and K2-functors modeled on Chevalley groups (to appear).Google Scholar
  94. 94.
    Ree, R.: A family of simple groups associated with the simple Lie algebras of type (G2), Amer. J. Math. 83 (1961), 401–420.Google Scholar
  95. 95.
    Ree, R.: A family of simple groups associated with the simple Lie algebras of type (G2), Amer. J. Math. 83 (1961), 432–462.Google Scholar
  96. 96.
    Ree, R.: Construction of certain semi-simple groups, Canad. J. Math. 16 (1964), 490–508.Google Scholar
  97. 97.
    Ringel, C. M.: Hall polynomials for the representation-finite hereditary algebras, Adv. Math. 84 (1990), 137–178.Google Scholar
  98. 98.
    Segal, G.: Unitary representations of some infinite dimensional groups, Comm. Math. Phys. 80 (1981), 301–342.Google Scholar
  99. 99.
    Seligman, G.: Modular Lie Algebras, Springer, Berlin, 1967.Google Scholar
  100. 100.
    Serre, J.-P.: Lie Algebras and Lie Groups, Benjamin, New York, 1965.Google Scholar
  101. 101.
    Seshadri, C. S.: Geometry of G/P. I. Standard Monomial Theory for Minuscule P, C. P. Ramanujan: a tribute, Tata Press, Bombay, 1978, pp. 207–239.Google Scholar
  102. 102.
    Sharpe, R. W.: On the structure of the Steinberg group St(Λ), J. Algebra 68(2) (1981), 453–467.Google Scholar
  103. 103.
    Splitthoff, S.: Finite presentability of Steinberg groups and related Chevalley groups, Contemp. Math. 55(2) (1986), 635–687.Google Scholar
  104. 104.
    Springer, T. A.: Linear Algebraic Groups, 2nd edn, Birkhäuser, Boston, 1981.Google Scholar
  105. 105.
    Springer, T. A.: Linear algebraic groups, Fundam. Trends in Math. 55 (1989), 5–136 (in Russian).Google Scholar
  106. 106.
    Stein, M. R.: Relativizing functors on rings and algebraic K-theory, J. Algebra 19(1) (1971), 140–152.Google Scholar
  107. 107.
    Stein, M. R.: Generators, relations and coverings of Chevalley groups over commutative rings, Amer. J. Math. 93(4) (1971), 965–1004.Google Scholar
  108. 108.
    Stein, M. R.: Surjective stability in dimension 0 for K2 and related functors, Trans. Amer. Math. Soc. 178(1) (1973), 165–191.Google Scholar
  109. 109.
    Stein, M. R.: Matsumoto's solution of the congruence subgroup problem and stability theorems in algebraic K-theory, in Proc. 19th Meeting Algebra Section Math. Soc. Japan, 1983, pp. 32–44.Google Scholar
  110. 110.
    Stein, M. R.: Stability theorems for K1, K2 and related functors modeled on Chevalley groups, Japan. J. Math. 4(1) (1978), 77–108.Google Scholar
  111. 111.
    Steinberg, R.: Genêrateurs, relations et revêtements des groupes algébriques, in Colloque sur la théorie des groupes algébriques (Bruxelles, 1962), Gauthier-Villars, Paris, 1962, pp. 113–127.Google Scholar
  112. 112.
    Steinberg, R.: Lectures on Chevalley Groups, Yale University, 1968.Google Scholar
  113. 113.
    Steinberg, R.: Some consequences of the elementary relations in SLn,Contemp. Math. 45 (1985), 335–350.Google Scholar
  114. 114.
    Stepanov, A. V.: Stability conditions in the theory of linear groups over rings, PhD Thesis, Leningrad State Univ., 1987 (in Russian).Google Scholar
  115. 115.
    Stepanov, A. V. and Vavilov, N. A.: Decomposition of transvections (to appear).Google Scholar
  116. 116.
    Stewart, I.: Central simplicity and Chevalley algebras, Compositio Math. 26(1) (1973), 111–118.Google Scholar
  117. 117.
    Sullivan, J. B.: Simply connected groups, the hyperalgebra and Verma conjectures, Amer. J. Math. 100(5) (1978), 1015–1031.Google Scholar
  118. 118.
    Suslin, A. A.: On a theorem of Cohn, J. Soviet Math. 17(2) (1981), 1801–1803.Google Scholar
  119. 119.
    Suslin, A. A.: On the structure of general linear group over polynomial ring, Soviet Math. Izv. 41(2) (1977), 503–516.Google Scholar
  120. 120.
    Suslin, A. A.: Algebraic K-theory, J. Soviet Math. 28(6) (1985), 870–923.Google Scholar
  121. 121.
    Suslin, A. A. and Kopeiko, V. I.: Quadratic modules and orthogonal groups over polynomial rings, J. Soviet Math. 20(6) (1982), 2665–2691.Google Scholar
  122. 122.
    Swan, R.: Generators and relations for certain special linear groups, Adv. Math. 6 (1971), 1–77.Google Scholar
  123. 123.
    Taddei, G.: Invariance du sous-groupe symplectique élémentaire dans le groupe symplectique sur un anneau, C.R. Acad. Sci. Paris, Sér. I 295(2) (1982), 47–50.Google Scholar
  124. 124.
    Taddei, G.: Schémas de Chevalley-Demazure, fonctions représentatives et thérème de normalité, Thèse, Univ. de Genève, 1985.Google Scholar
  125. 125.
    Taddei, G.: Normalité des groupes élémentaire dans les groupes de Chevalley sur un anneau, Contemp. Math. 55(II) (1986), 693–710.Google Scholar
  126. 126.
    Tits, J.: Sur les constantes de structure et le théorème d'existence des algèbres de Lie semisimples. Publ. Math. Inst. Hautes Et. Sci. (1966), No. 31, pp. 21–58.Google Scholar
  127. 127.
    Tits, J.: Normalisateurs de tores. I. Groupes de Coxeter étendus, J. Algebra 4(1) (1966), 96–116.Google Scholar
  128. 128.
    Tulenbaev, M. S.: The Schur multiplier of the group of elementary matrices of finite order, J. Soviet Math. 17(4) (1981), 2062–2067.Google Scholar
  129. 129.
    Vaserstein, L. N.: On the stabilization of the general linear group over a ring, Math. USSR Sb. 8 (1969), 383–400.Google Scholar
  130. 130.
    Vaserstein, L. N. Stabilization of unitary and orthogonal groups over a ring, Math. USSR Sb. 10 (1970), 307–326.Google Scholar
  131. 131.
    Vaserstein, L. N. The stable rank of rings and the dimension of topological spaces, Funct. Anal. Appl. 5 (1971), 102–110.Google Scholar
  132. 132.
    Vaserstein, L. N. Stabilization for the classical groups over rings, Math. USSR Sb. 22 (1974), 271–303.Google Scholar
  133. 133.
    Vaserstein, L. N. On normal subgroups of GLnover a ring, in Lecture Notes in Math. 854, Springer, New York, 1981, pp. 456–465.Google Scholar
  134. 134.
    Vaserstein, L. N. On normal subgroups of Chevalley groups over commutative rings, Tôhoku Math. J. 38 (1986), 219–230.Google Scholar
  135. 135.
    Vaserstein, L. N. Normal subgroups of orthogonal groups over commutative rings, Amer. J. Math. 110(5) (1988), 955–973.Google Scholar
  136. 136.
    Vaserstein, L. N. Normal subgroups of symplectic groups over rings, K-Theory 2(5) (1989), 647–673.Google Scholar
  137. 137.
    Vavilov, N. A.: Parabolic subgroups of Chevalley groups over a commutative ring, J. Soviet Math. 26(3) (1984), 1848–1860.Google Scholar
  138. 138.
    Vavilov, N. A.: Subgroups of split classical groups, Dr. Sci. Thesis (Habilitations-schrift), Leningrad State Univ., 1987 (in Russian).Google Scholar
  139. 139.
    Vavilov, N. A.: Structure of split classical groups over commutative rings, Soviet Math. Dokl. 37 (1988), 550–553.Google Scholar
  140. 140.
    Vavilov, N. A.: On subgroups of the split classical groups, Proc. Math. Inst. Steklov (1991), No. 4, pp. 27–41.Google Scholar
  141. 141.
    Vavilov, N. A.: Structure of Chevalley groups over commutative rings, Proc. Conf. Nonassociative Algebras and Related Topics (Hiroshima, 1990), World Scientific, Singapore, 1991, pp. 219–335.Google Scholar
  142. 142.
    Vavilov, N. A.: Do it yourself structure constants for Lie algebras of type El,Preprint Universität Bielefeld, 1993.Google Scholar
  143. 143.
    Vavilov, N. A.: Intermediate subgroups in Chevalley groups, Proc. Conf. Groups of Lie Type and their Geometrics, Cambridge Univ. Press, London Math. Soc. Lec. Notes 1207 (1995), 233–280.Google Scholar
  144. 144.
    Vavilov, N. A.: Linear groups over general rings. I, II (to appear).Google Scholar
  145. 145.
    Vavilov, N. A.: Weight elements of Chevalley groups (to appear).Google Scholar
  146. 146.
    Vavilov, N. A., Plotkin, E. B., and Stepanov, A. V.: Calculations in Chevalley groups over commutative rings, Soviet Math. Dokl. 40(1) (1990), 145–147.Google Scholar
  147. 147.
    Vinberg, E. and Onishchik, E.: Lie Groups and Algebraic Groups, Springer, Berlin, 1988.Google Scholar
  148. 148.
    Vorst, T.: The general linear group of polynomial rings over regular rings, Comm. Algebra 9(5) (1981), 499–509.Google Scholar
  149. 149.
    Waterhouse, W. C.: Introduction to Affine Group Schemes, Springer, Berlin, 1979.Google Scholar
  150. 150.
    Waterhouse, W. C.: Automorphisms of GLn(R), Proc. Amer. Math. Soc. 79 (1980), 347–351.Google Scholar
  151. 151.
    Wagner, A.: On the classification of the classical groups, Math. Z. 97(1) (1967), 66–76.Google Scholar
  152. 152.
    Wenzel, C.: Rationality of G/P for a nonreduced parabolic subgroup scheme P, Proc. Amer. Math. Soc. 117 (1993), 899–904.Google Scholar
  153. 153.
    Wenzel, C.: Classification of all parabolic subgroup schemes of a reductive linear algebraic group over an algebraic closed field, Trans. Amer. Math. Soc. 337 (1993), 211–218.Google Scholar
  154. 154.
    Wilson, J. S.: The normal and subnormal structure of general linear groups, Proc. Cambridge Philos. Soc. 71 (1972), 163–177.Google Scholar
  155. 155.
    Zalesskii, A. E.: Linear groups, J. Soviet Math. 31(3) (1985), 2974–3004.Google Scholar
  156. 156.
    Zelmanov, E. I.: Isomorphisms of linear groups over an associative rings, Siberian Math. J. 26(4) (1986), 515–530.Google Scholar
  157. 157.
    Suslin, A. A. and Tulenbaev, M. S.: Stabilization theorem for the Milnor K 2 functor, J. Soviet Math. 17 (1981), 1804–1819.Google Scholar
  158. 158.
    Dennis, R. K.: Stability for K 2, in Lecture Notes in Math. 353, Springer, New York, 1973, pp. 85–94.Google Scholar
  159. 159.
    Dennis, R. K.: K 2 and the atable range condition, Manuscript, Princeton, 1971, 28 pp.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Nikolai Vavilov
    • 1
    • 2
  • Eugene Plotkin
    • 3
  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  2. 2.Department of Mathematics and MechanicsUniversity of Sant PetersburgPetrodvoretsRussia
  3. 3.Department of Mathematics and Computer ScienceBar Ilan UniversityRamat GanIsrael

Personalised recommendations