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Overgroups of elementary symplectic groups


Authors: N. A. Vavilov and V. A. Petrov
Translated by: the authors
Original publication: Algebra i Analiz, tom 15 (2003), nomer 4.
Journal: St. Petersburg Math. J. 15 (2004), 515-543
MSC (2000): Primary 20G35
DOI: https://doi.org/10.1090/S1061-0022-04-00820-9
Published electronically: July 6, 2004
MathSciNet review: 2068980
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Abstract: Let $R$ be a commutative ring, and let $l\ge 2$; for $l=2$ it is assumed additionally that $R$ has no residue fields of two elements. The subgroups of the general linear group $\operatorname{GL}(n,R)$ that contain the elementary symplectic group $\operatorname{Ep}(2l,R)$ are described. In the case where $R=K$ is a field, similar results were obtained earlier by Dye, King, and Shang Zhi Li.


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  • 1. Hyman Bass, Algebraic 𝐾-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
  • 2. H. Bass, J. Milnor, and J.-P. Serre, Solution of the congruence subgroup problem for 𝑆𝐿_{𝑛}(𝑛≥3) and 𝑆𝑝_{2𝑛}(𝑛≥2), Inst. Hautes Études Sci. Publ. Math. 33 (1967), 59–137. MR 0244257
  • 3. E. L. Bashkirov, Linear groups containing the special unitary group of non-zero index, Vestsi Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk 1986, no. 5, 120-121 (complete text of manuscript sits at VINITI 7.08.1985, no. 5897-85 Dep.). (Russian)
  • 4. -, Linear groups containing the symplectic group, Vestsi Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk 1987, no. 3, 116-117 (complete text of manuscript sits at VINITI 11.04.1986, no. 2616-B86 Dep.). (Russian)
  • 5. E. L. Bashkirov, Linear groups that contain the group 𝑆𝑝_{𝑛}(𝑘) over a field of characteristic 2, Vestsī Akad. Navuk BSSR Ser. Fīz.-Mat. Navuk 4 (1991), 21–26, 123 (Russian, with English summary). MR 1141150
  • 6. E. L. Bashkirov, Linear groups that contain the commutant of an orthogonal group of index greater than 1, Sibirsk. Mat. Zh. 33 (1992), no. 5, 15–21, 221 (Russian, with Russian summary); English transl., Siberian Math. J. 33 (1992), no. 5, 754–759 (1993). MR 1197069, https://doi.org/10.1007/BF00970984
  • 7. E. L. Bashkirov, On subgroups of the general linear group over the skew field of quaternions containing the special unitary group, Sibirsk. Mat. Zh. 39 (1998), no. 6, 1251–1266, i (Russian, with Russian summary); English transl., Siberian Math. J. 39 (1998), no. 6, 1080–1092. MR 1672621, https://doi.org/10.1007/BF02674119
  • 8. Z. I. Borevich and N. A. Vavilov, Arrangement of subgroups in the general linear group over a commutative ring, Trudy Mat. Inst. Steklov. 165 (1984), 24-42; English transl. in Proc. Steklov Inst. Math. 1985, no. 3. MR 0752930 (86e:20052)
  • 9. N. A. Vavilov, Subgroups of split classical groups, Doctor's Thesis, Leningrad. Gos. Univ., Leningrad, 1987. (Russian). MR 0953017 (89g:20074)
  • 10. -, The structure of split classical groups over a commutative ring, Dokl. Akad. Nauk SSSR 299 (1988), no. 6, 1300-1303; English transl., Soviet Math. Dokl. 37 (1988), no. 2, 550-553. MR 0947412 (89j:20053)
  • 11. -, Subgroups of split orthogonal groups over a ring, Sibirsk. Mat. Zh. 29 (1988), no. 4, 31-43; English transl., Siberian Math. J. 29 (1988), no. 4, 537-547 (1989). MR 0969101 (90c:20061)
  • 12. N. A. Vavilov, Subgroups of splittable classical groups, Trudy Mat. Inst. Steklov. 183 (1990), 29–42, 223 (Russian). Translated in Proc. Steklov Inst. Math. 1991, no. 4, 27–41; Galois theory, rings, algebraic groups and their applications (Russian). MR 1092012
  • 13. N. A. Vavilov, On subgroups of the general symplectic group over a commutative ring, Rings and modules. Limit theorems of probability theory, No. 3 (Russian), Izd. St.-Peterbg. Univ., St. Petersburg, 1993, pp. 16–38, 256 (Russian, with Russian summary). MR 1351048
  • 14. N. A. Vavilov, Subgroups of split orthogonal groups over a commutative ring, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 281 (2001), no. Vopr. Teor. Predst. Algebr. i Grupp. 8, 35–59, 280 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N. Y.) 120 (2004), no. 4, 1501–1512. MR 1875717, https://doi.org/10.1023/B:JOTH.0000017881.22871.49
  • 15. N. A. Vavilov and E. V. Dybkova, Subgroups of the general symplectic group containing the group of diagonal matrices, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 103 (1980), 31-47; English transl. in J. Soviet Math. 24 (1984), no. 4. MR 0618492 (82h:20054)
  • 16. N. A. Vavilov and V. A. Petrov, On supergroups of 𝐸𝑂(2𝑙,𝑅), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 272 (2000), no. Vopr. Teor. Predst. Algebr i Grupp. 7, 68–85, 345–346 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N. Y.) 116 (2003), no. 1, 2917–2925. MR 1811793, https://doi.org/10.1023/A:1023442407926
  • 17. L. N. Vaseršteĭn, On the stabilization of the general linear group over a ring, Math. USSR-Sb. 8 (1969), 383–400. MR 0267009
  • 18. L. N. Vaseršteĭn, Stabilization of unitary and orthogonal groups over a ring with involution, Mat. Sb. (N.S.) 81 (123) (1970), 328–351 (Russian). MR 0269722
  • 19. L. N. Vaseršteĭn and A. A. Suslin, Serre’s problem on projective modules over polynomial rings, and algebraic 𝐾-theory, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 5, 993–1054, 1199 (Russian). MR 0447245
  • 20. I. Z. Golubchik, Normal subgroups of the orthogonal group over the associative ring with involution, Uspekhi Mat. Nauk 30 (1975), no. 6, 165. (Russian)
  • 21. -, Subgroups of the general linear $\operatorname{GL}_n(R)$ over an associative ring $R$, Uspekhi Mat. Nauk 39 (1984), no. 1, 125-126; English transl. in Russian Math. Surveys 39 (1984), no. 1. MR 0733962 (85j:20042)
  • 22. -, Normal subgroups of the linear and unitary groups over associative rings, Spaces over Algebras, and Some Problems in the Theory of Nets, Bashkir. Gos. Ped. Inst., Ufa, 1985, pp. 122-142. (Russian) MR 0975035
  • 23. V. I. Kopeiko, Stabilization of symplectic groups over a ring of polynomials, Mat. Sb. (N. S.) 106 (1978), no. 1, 94-107; English transl., Math. USSR-Sb. 34 (1978), no. 5, 655-669. MR 0497932 (80f:13008)
  • 24. O. T. O'Meara, Lectures on symplectic groups, Univ. Notre Dame, 1976.
  • 25. Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR 0466335
  • 26. A. V. Stepanov, Stability conditions in the theory of linear groups over rings, Ph.D. Thesis, Leningrad. Gos. Univ., Leningrad, 1987. (Russian)
  • 27. A. V. Stepanov, On the arrangement of subgroups normalized by a fixed subgroup, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 198 (1991), no. Voprosy Teor. Predstav. Algebr Grupp. 2, 92–102, 113 (Russian); English transl., J. Soviet Math. 64 (1993), no. 1, 769–776. MR 1164862, https://doi.org/10.1007/BF02988482
  • 28. A. A. Suslin, The structure of the special linear group over rings of polynomials, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 2, 235–252, 477 (Russian). MR 0472792
  • 29. E. Abe, Whitehead groups of Chevalley groups over polynomial rings, Comm. Algebra 11 (1983), no. 12, 1271-1307. MR 0697617 (85d:20038)
  • 30. -, Chevalley groups over commutative rings, Radical Theory (Sendai, 1988), Uchida Rokakuho, Tokyo, 1989, pp. 1-23. MR 0999577 (91a:20047)
  • 31. -, Normal subgroups of Chevalley groups over commutative rings, Algebraic $K$-Theory and Algebraic Number Theory (Honolulu, HI, 1987), Contemp. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1989. MR 0991973 (91a:20046)
  • 32. Eiichi Abe and Kazuo Suzuki, On normal subgroups of Chevalley groups over commutative rings, Tôhoku Math. J. (2) 28 (1976), no. 2, 185–198. MR 0439947, https://doi.org/10.2748/tmj/1178240833
  • 33. M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), no. 3, 469-514. MR 0746539 (86a:20054)
  • 34. A. Bak, The stable structure of quadratic modules, Thesis, Columbia Univ., 1969.
  • 35. -, $K$-theory of forms, Ann. of Math. Stud., vol. 98, Princeton Univ. Press, Princeton, NJ, 1981. MR 0632404 (84m:10012)
  • 36. Anthony Bak, Nonabelian 𝐾-theory: the nilpotent class of 𝐾₁ and general stability, 𝐾-Theory 4 (1991), no. 4, 363–397. MR 1115826, https://doi.org/10.1007/BF00533991
  • 37. Anthony Bak and Alexei Stepanov, Dimension theory and nonstable 𝐾-theory for net groups, Rend. Sem. Mat. Univ. Padova 106 (2001), 207–253. MR 1876221
  • 38. Anthony Bak and Nikolai Vavilov, Normality for elementary subgroup functors, Math. Proc. Cambridge Philos. Soc. 118 (1995), no. 1, 35–47. MR 1329456, https://doi.org/10.1017/S0305004100073436
  • 39. Anthony Bak and Nikolai Vavilov, Structure of hyperbolic unitary groups. I. Elementary subgroups, Algebra Colloq. 7 (2000), no. 2, 159–196. MR 1810843, https://doi.org/10.1007/s100110050017
  • 40. -, Structure of hyperbolic unitary groups. . Normal subgroups (to appear).
  • 41. Hyman Bass, Unitary algebraic 𝐾-theory, Algebraic K-theory, III: Hermitian K-theory and geometric applications (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 57–265. Lecture Notes in Math., Vol. 343. MR 0371994
  • 42. Roger W. Carter, Simple groups of Lie type, John Wiley & Sons, London-New York-Sydney, 1972. Pure and Applied Mathematics, Vol. 28. MR 0407163
  • 43. Douglas L. Costa and Gordon E. Keller, The 𝐸(2,𝐴) sections of 𝑆𝐿(2,𝐴), Ann. of Math. (2) 134 (1991), no. 1, 159–188. MR 1114610, https://doi.org/10.2307/2944335
  • 44. Douglas L. Costa and Gordon E. Keller, Radix redux: normal subgroups of symplectic groups, J. Reine Angew. Math. 427 (1992), 51–105. MR 1162432
  • 45. Lino Di Martino and Nikolai Vavilov, (2,3)-generation of 𝑆𝐿(𝑛,𝑞). I. Cases 𝑛=5,6,7, Comm. Algebra 22 (1994), no. 4, 1321–1347. MR 1261262, https://doi.org/10.1080/00927879408824908
  • 46. R. H. Dye, Interrelations of symplectic and orthogonal groups in characteristic two, J. Algebra 59 (1979), no. 1, 202-221. MR 0541675 (81c:20028)
  • 47. -, On the maximality of the orthogonal groups in the symplectic groups in characteristic two, Math. Z. 172 (1980), no. 3, 203-212. MR 0581439 (81h:20060)
  • 48. -, Maximal subgroups of $\operatorname{GL}_{2n}(K)$, $\operatorname{SL}_{2n}(K)$, $\operatorname{PGL}_{2n}(K)$, and $\operatorname{PSL}_{2n}(K)$ associated with symplectic polarities, J. Algebra 66 (1980), no. 1, 1-11. MR 0591244 (81j:20061)
  • 49. Fritz Grunewald, Jens Mennicke, and Leonid Vaserstein, On symplectic groups over polynomial rings, Math. Z. 206 (1991), no. 1, 35–56. MR 1086811, https://doi.org/10.1007/BF02571323
  • 50. Alexander J. Hahn and O. Timothy O’Meara, The classical groups and 𝐾-theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 291, Springer-Verlag, Berlin, 1989. With a foreword by J. Dieudonné. MR 1007302
  • 51. Roozbeh Hazrat, Dimension theory and nonstable 𝐾₁ of quadratic modules, 𝐾-Theory 27 (2002), no. 4, 293–328. MR 1962906, https://doi.org/10.1023/A:1022623004336
  • 52. Roozbeh Hazrat and Nikolai Vavilov, 𝐾₁ of Chevalley groups are nilpotent, J. Pure Appl. Algebra 179 (2003), no. 1-2, 99–116. MR 1958377, https://doi.org/10.1016/S0022-4049(02)00292-X
  • 53. Wolfram Jehne, Die Struktur der symplektischen Gruppe über lokalen und dedekindschen Ringen, S.-B. Heidelberger Akad. Wiss. Math.-Natur. Kl. 1962/64 (1962/1964), 187–235 (German). MR 0175999
  • 54. O. H. King, On subgroups of the special linear group containing the special orthogonal group, J. Algebra 96 (1985), no. 1, 178-193. MR 0808847 (87b:20057)
  • 55. -, On subgroups of the special linear group containing the special unitary group, Geom. Dedicata 19 (1985), no. 3, 297-310. MR 0815209 (87c:20081)
  • 56. Oliver King, Subgroups of the special linear group containing the diagonal subgroup, J. Algebra 132 (1990), no. 1, 198–204. MR 1060843, https://doi.org/10.1016/0021-8693(90)90263-N
  • 57. Franz Kirchheimer, Die Normalteiler der symplektischen Gruppen über beliebigen lokalen Ringen, J. Algebra 50 (1978), no. 1, 228–241 (German). MR 0578492, https://doi.org/10.1016/0021-8693(78)90184-9
  • 58. Peter Kleidman and Martin Liebeck, The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series, vol. 129, Cambridge University Press, Cambridge, 1990. MR 1057341
  • 59. Wilhelm Klingenberg, Symplectic groups over local rings, Amer. J. Math. 85 (1963), 232–240. MR 0153749, https://doi.org/10.2307/2373212
  • 60. Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tignol, The book of involutions, American Mathematical Society Colloquium Publications, vol. 44, American Mathematical Society, Providence, RI, 1998. With a preface in French by J. Tits. MR 1632779
  • 61. Fu An Li, The structure of symplectic groups over arbitrary commutative rings, Acta Math. Sinica (N.S.) 3 (1987), no. 3, 247-255. MR 0916269 (88m:20098)
  • 62. Fu An Li, The structure of orthogonal groups over arbitrary commutative rings, Chinese Ann. Math. Ser. B 10 (1989), no. 3, 341–350. A Chinese summary appears in Chinese Ann. Math. Ser. A 10 (1989), no. 4, 520. MR 1027673
  • 63. Shang Zhi Li, Overgroups of 𝑆𝑈(𝑛,𝐾,𝑓) or Ω(𝑛,𝐾,𝑄) in 𝐺𝐿(𝑛,𝐾), Geom. Dedicata 33 (1990), no. 3, 241–250. MR 1050412, https://doi.org/10.1007/BF00181331
  • 64. Shang Zhi Li, Overgroups of a unitary group in 𝐺𝐿(2,𝐾), J. Algebra 149 (1992), no. 2, 275–286. MR 1172429, https://doi.org/10.1016/0021-8693(92)90016-F
  • 65. Shang Zhi Li, Overgroups in 𝐺𝐿(𝑛,𝐹) of a classical group over a subfield of 𝐹, Algebra Colloq. 1 (1994), no. 4, 335–346. MR 1301157
  • 66. Shang Zhi Li and Zong Li Wei, Overgroups of a symplectic group in a linear group over a Euclidean ring, J. Univ. Sci. Technol. China 32 (2002), no. 2, 127–134 (Chinese, with English and Chinese summaries). MR 1911131
  • 67. M. Newman, Matrix completion theorems, Proc. Amer. Math. Soc. 94 (1985), no. 1, 39-45. MR 0781052 (86d:15009)
  • 68. V. A. Petrov, On the first cohomology of unitary Steinberg groups (to appear). (English)
  • 69. Viktor Petrov, Overgroups of unitary groups, 𝐾-Theory 29 (2003), no. 3, 147–174. MR 2028500, https://doi.org/10.1023/B:KTHE.0000006934.95243.91
  • 70. Michael R. Stein, Generators, relations and coverings of Chevalley groups over commutative rings, Amer. J. Math. 93 (1971), 965–1004. MR 0322073, https://doi.org/10.2307/2373742
  • 71. -, Stability theorems for $\operatorname{K}_1$, $\operatorname{K}_2$ and related functors modeled on Chevalley groups, Japan. J. Math. (N.S.) 4 (1978), no. 1, 77-108. MR 0528869 (81c:20031)
  • 72. A. Stepanov, Non-standard subgroups between $E(n,R)$ and $\operatorname{GL}(n,A)$, Algebra Colloq. (to appear).
  • 73. Alexei Stepanov and Nikolai Vavilov, Decomposition of transvections: a theme with variations, 𝐾-Theory 19 (2000), no. 2, 109–153. MR 1740757, https://doi.org/10.1023/A:1007853629389
  • 74. G. Taddei, Invariance du sous-groupe symplectique élémentaire dans le groupe symplectique sur un anneau, C. R. Acad. Sci Paris Sér I Math. 295 (1982), no. 2, 47-50. MR 0676359 (84c:20058)
  • 75. -, Normalité des groupes élémentaires dans les groupes de Chevalley sur un anneau, Applications of Algebraic $K$-Theory to Algebraic Geometry and Number Theory, Part I, II (Boulder, Colo., 1983), Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 693-710. MR 0862660 (88a:20054)
  • 76. F. G. Timmesfeld, Abstract root subgroups and quadratic action, Adv. Math. 142 (1999), no. 1, 1–150. With an appendix by A. E. Zalesskii. MR 1671440, https://doi.org/10.1006/aima.1998.1779
  • 77. Jacques Tits, Systèmes générateurs de groupes de congruence, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 9, Ai, A693–A695 (French, with English summary). MR 0424966
  • 78. L. N. Vaserstein, On the normal subgroups of the $\operatorname{GL}_n$ over a ring, Algebraic $K$-Theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980), Lecture Notes in Math., vol. 854, Springer, Berlin-New York, 1981, pp. 454-465. MR 0618316 (83c:20058)
  • 79. -, An answer to a question of M. Newman on matrix completion, Proc. Amer. Math. Soc. 97 (1986), no. 2, 189-196. MR 0835863 (88e:18012)
  • 80. -, On normal subgroups of Chevalley groups over commutative rings, Tôhoku Math. J. 38 (1986), no. 2, 219-230. MR 0843808 (87k:20081)
  • 81. -, Normal subgroups of orthogonal groups over commutative rings, Amer. J. Math. 110 (1988), no. 5, 955-973. MR 0961501 (89i:20071)
  • 82. -, Normal subgroups of symplectic groups over rings, Proceedings of Research Symposium on $K$-Theory and its Applications (Ibadan, 1987), $K$-Theory 2 (1989), no. 5, 647-673. MR 0999398 (90f:20064)
  • 83. Leonid N. Vaserstein and Hong You, Normal subgroups of classical groups over rings, J. Pure Appl. Algebra 105 (1995), no. 1, 93–105. MR 1364152, https://doi.org/10.1016/0022-4049(94)00144-8
  • 84. Nikolai A. Vavilov, Structure of Chevalley groups over commutative rings, Nonassociative algebras and related topics (Hiroshima, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 219–335. MR 1150262
  • 85. Nikolai Vavilov, Intermediate subgroups in Chevalley groups, Groups of Lie type and their geometries (Como, 1993) London Math. Soc. Lecture Note Ser., vol. 207, Cambridge Univ. Press, Cambridge, 1995, pp. 233–280. MR 1320525, https://doi.org/10.1017/CBO9780511565823.018
  • 86. -, The work of Borevich on linear groups, and beyond, Proc. Internat. Algebra Conf. (St. Petersburg, 2002), Marcel Dekker, 2003 (to appear).
  • 87. J. S. Wilson, The normal and subnormal structure of general linear groups, Proc. Cambridge Philos. Soc. 71 (1972), 163–177. MR 0291304
  • 88. Hong You and Baodong Zheng, Overgroups of symplectic group in linear group over local rings, Comm. Algebra 29 (2001), no. 6, 2313–2318. MR 1845112, https://doi.org/10.1081/AGB-100002390
  • 89. E. V. Dybkova, Overdiagonal subgroups of the hyperbolic unitary group for a good form ring over a field, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 236 (1997), no. Vopr. Teor. Predst. Algebr i Grupp. 5, 87–96, 216–217 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 95 (1999), no. 2, 2096–2101. MR 1754447, https://doi.org/10.1007/BF02169965
  • 90. E. V. Dybkova, On overdiagonal subgroups of the hyperbolic unitary group over a noncommutative skew field, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 289 (2002), no. Vopr. Teor. Predst. Algebr. i Grupp. 9, 154–206, 302 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 124 (2004), no. 1, 4766–4791. MR 1949740, https://doi.org/10.1023/B:JOTH.0000042313.09130.b6

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Additional Information

N. A. Vavilov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospekt 28, St. Petersburg, 198504, Russia

V. A. Petrov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospekt 28, St. Petersburg 198504, Russia

DOI: https://doi.org/10.1090/S1061-0022-04-00820-9
Received by editor(s): February 18, 2003
Published electronically: July 6, 2004
Additional Notes: The present paper has been written in the framework of the RFBR projects nos. 01-01-00924 and 00-01-00441, and INTAS 00-566. The theorem on decomposition of unipotents mentioned in §13 is a part of first author’s joint work with A. Bak and was carried out at the University of Bielefeld with the support of AvH-Stiftung, SFB-343, and INTAS 93-436. At the final stage, the work of the authors was supported by express grants of the Russian Ministry of Higher Education ‘Geometry of root subgroups’ PD02-1.1-371 and ‘Overgroups of semisimple groups’ E02-1.0-61.
Article copyright: © Copyright 2004 American Mathematical Society

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