St. Petersburg Mathematical Society

An $\mathrm{A}_2$ -proof of structure theorems for Chevalley groups of types $\mathrm{E}_6$ and $\mathrm{E}_7$

Author(s): N. A. Vavilov; M. R. Gavrilovich
Translated by: N. A. Vavilov
Original publication: Algebra i Analiz, tom 16 (2004), vypusk 4.
Journal: St. Petersburg Math. J. 16 (2005), 649-672.
MSC (2000): Primary 20G15, 20G35
Posted: June 21, 2005
Retrieve article in: PDF DVI PostScript

1.: H. Bass, Algebraic -theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491 (40:2736)
2.: H. Bass, J. Milnor, and J.-P. Serre, Solution of the congruence subgroup problem for ${SL}_n$ $(n\ge 3)$ and ${Sp}_{2n}$ $(n\ge 2)$ , Publ. Math. No. 33 (1967), 59-137. MR 0244257 (39:5574)
3.: 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)
4.: A. Borel, Properties and linear representations on Chevalley groups, Seminar on Algebraic Groups and Related Finite Groups (Princeton, NJ, 1968/69), Lecture Notes in Math., vol. 131, Springer-Verlag, Berlin-New York, 1970, pp. A1-A55. MR 0258838 (41:3484)
5.: N. Bourbaki, Eléments de mathématique. Fasc. XXVI, XXXVII. Groupes et algèbres de Lie. Chapitres I-III, Actualités Sci. Indust., No. 1285, 1349, Hermann, Paris, 1971, 1972. MR 0271276 (42:6159); MR 0573068 (58:28083a)
6.: -, Eléments de mathématique. Fasc. XXXVIII. Groupes et algèbres de Lie. Chapitres VII, VIII, Actualités Sci. Indust., No. 1364, Hermann, Paris, 1975. MR 0453824 (56:12077)
7.: N. A. Vavilov, Parabolic subgroups of the Chevalley group over a commutative ring, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 116 (1982), 20-43; English transl. in J. Soviet Math. 26 (1984), no. 3. MR 0687837 (85j:20046a)
8.: -, Subgroups of split classical groups, Doctor's Thesis, Leningrad. Gos. Univ., Leningrad, 1987. (Russian)
9.: -, 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)
10.: -, 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)
11.: -, Subgroups of split classical groups, Trudy Mat. Inst. Steklov. 183 (1990), 29-42; English transl., Proc. Steklov Inst. Math. 1991, no. 4, 27-41. MR 1092012 (92d:20065)
12.: -, On subgroups of the general symplectic group over a commutative ring, Rings and Modules. Limit Theorems of Probability Theory, No. 3, S.-Peterburg. Univ., St. Petersburg, 1993, pp. 16-38. (Russian) MR 1351048 (96j:20066)
13.: -, Subgroups of split orthogonal groups over a commutative ring, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 281 (2001), 35-59; English transl., J. Math. Sci. 120 (2004), no. 4, 1501-1512. MR 1875717 (2002j:20099)
14.: -, How to see the signs of structure constants (in preparation).
15.: N. A. Vavilov, E. B. Plotkin, and A. V. Stepanov, Calculations in Chevalley groups over commutative rings, Dokl. Akad. Nauk SSSR 307 (1989), no. 4, 788-791; English transl., Soviet Math. Dokl. 40 (1990), no. 1, 145-147. MR 1020667 (90j:20093)
16.: L. N. Vasershtein and A. A. Suslin, Serre's problem on projective modules over polynomial rings, and algebraic -theory, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 5, 993-1054; English transl. in Math. USSR-Izv. 10 (1976), no. 5. MR 0447245 (56:5560)
17.: I. Z. Golubchik, The full linear group over an associative ring, Uspekhi Mat. Nauk 28 (1973), no. 3, 179-180. (Russian) MR 0396783 (53:643)
18.: -, On normal subgroups of the orthogonal group over an associative ring with involution, Uspekhi Mat. Nauk 30 (1975), no. 6, 165. (Russian)
19.: -, 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
20.: 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)
21.: R. Steinberg, Lectures on Chevalley groups, Yale Univ., New Haven, Conn., 1968. MR 0466335 (57:6215)
22.: A. V. Stepanov, Stability conditions in the theory of linear groups over rings, Ph. D. Thesis, Leningrad. Gos. Univ., Leningrad, 1987. (Russian)
23.: 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; English transl. in Math. USSR-Izv. 11 (1977), no. 2. MR 0472792 (57:12482)
24.: A. A. Suslin and V. I. Kopeiko, Quadratic modules and the orthogonal group over polynomial rings, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 71 (1977), 216-250; English transl. in J. Soviet Math. 20 (1982), no. 6. MR 0469914 (57:9694)
25.: E. Abe, Chevalley groups over local rings, Tôhoku Math. J. (2) 21 (1969), 474-494. MR 0258837 (41:3483)
26.: -, Whitehead groups of Chevalley groups over polynomial rings, Comm. Algebra 11 (1983), 1271-1307. MR 0697617 (85d:20038)
27.: -, Chevalley groups over commutative rings, Radical Theory (Sendai, 1988), Uchida Rokakuho, Tokyo, 1989, pp. 1-23. MR 0999577 (91a:20047)
28.: -, Normal subgroups of Chevalley groups over commutative rings, Algebraic -Theory and Algebraic Number Theory (Honolulu, HI, 1987), Contemp. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1989, pp. 1-17. MR 0991973 (91a:20046)
29.: E. Abe and J. Hurley, Centers of Chevalley groups over commutative rings, Comm. Algebra 16 (1988), 57-74. MR 0921942 (90e:20040)
30.: E. Abe and K. Suzuki, On normal subgroups of Chevalley groups over commutative rings, Tôhoku Math. J. (2) 28 (1976), 185-198. MR 0439947 (55:12828)
31.: H. Azad, M. Barry, and G. M. Seitz, On the structure of parabolic subgroups, Comm. Algebra 18 (1990), 551-562. MR 1047327 (91d:20048)
32.: A. Bak, The stable structure of quadratic modules, Thesis, Columbia Univ., 1969.
33.: A. Bak and N. Vavilov, Normality of elementary subgroup functors, Math. Proc. Cambridge Philos. Soc. 118 (1995), 35-47. MR 1329456 (96d:20046)
34.: -, Structure of hyperbolic unitary groups. I. Elementary subgroups, Algebra Colloq. 7 (2000), 159-196. MR 1810843 (2002b:20070)
35.: H. Bass, -theory and stable algebra, Inst. Hautes Études Sci. Publ. Math. No. 22 (1964), 5-60. MR 0174604 (30:4805)
36.: -, Unitary algebraic -theory, Algebraic -Theory, III: Hermitian -Theory and Geometric Applications (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math., vol. 343, Springer, Berlin, 1973, pp. 57-265. MR 0371994 (51:8211)
37.: R. Carter, Simple groups of Lie type, Pure Appl. Math., vol. 28, John Wiley, London et al., 1972. MR 0407163 (53:10946)
38.: D. L. Costa and G. E. Keller, The sections of , Ann. of Math. (2) 134 (1991), 159-188. MR 1114610 (92f:20047)
39.: -, Radix redux: normal subgroups of symplectic groups, J. Reine Angew. Math. 427 (1992), 51-105. MR 1162432 (93h:20053)
40.: -, On the normal subgroups of $\G_2(A)$ , Trans. Amer. Math. Soc. 351 (1999), 5051-5088. MR 1487611 (2000c:20070)
41.: M. Gavrilovich and N. Vavilov, Structure of Chevalley groups: the proof from the Book (in preparation).
42.: A. J. Hahn and O. T. O'Meara, The classical groups and K-theory, Grundlehren Math. Wiss., vol. 291, Springer-Verlag, Berlin etc., 1989. MR 1007302 (90i:20002)
43.: R. Hazrat and N. A. Vavilov, $\mathrm{K}_1$ of Chevalley groups are nilpotent, J. Pure Appl. Algebra 179 (2003), 99-116. MR 1958377 (2004i:20081)
44.: Fu An Li, The structure of symplectic groups over arbitrary commutative rings, Acta Math. Sinica (N. S.) 3 (1987), 247-255. MR 0916269 (88m:20098)
45.: -, The structure of orthogonal groups over arbitrary commutative rings, Chinese Ann. Math. Ser. B 10 (1989), 341-350. MR 1027673 (90k:20084)
46.: M. W. Liebeck and G. M. Seitz, Subgroups generated by root elements in groups of Lie type, Ann. of Math. (2) 139 (1994), 293-361. MR 1274094 (95d:20078)
47.: H. Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Sup. (4) 2 (1969), 1-62. MR 0240214 (39:1566)
48.: E. B. Plotkin, On the stability of the $\mathrm{K}_1$ -functor for Chevalley groups of type ${E}_7$ , J. Algebra 210 (1998), 67-85. MR 1656415 (99k:20099)
49.: E. B. Plotkin, A. A. Semenov, and N. A. Vavilov, Visual basic representations: an atlas, Internat. J. Algebra Comput. 8 (1998), 61-95. MR 1492062 (98m:17010)
50.: R. Richardson, G. E. Röhrle, and R. Steinberg, Parabolic subgroups with abelian unipotent radical, Invent. Math. 110 (1992), 649-671. MR 1189494 (93j:20092)
51.: G. E. Röhrle, Orbits in internal Chevalley modules, Groups, Combinatorics and Geometry (Durham, 1990), London Math. Soc. Lecture Note Ser., vol. 165, Cambridge Univ. Press, Cambridge, 1992, pp. 311-315. MR 1200268
52.: -, On the structure of parabolic subgroups in algebraic groups, J. Algebra 157 (1993), 80-115. MR 1219660 (94d:20053)
53.: -, On extraspecial parabolic subgroups, Linear Algebraic Groups and their Representations (Los Angeles, CA, 1992), Contemp. Math., vol. 153, Amer. Math. Soc., Providence, RI, 1993, pp. 143-155. MR 1247502 (94k:20082)
54.: M. R. Stein, Generators, relations and coverings of Chevalley groups over commutative rings, Amer. J. Math. 93 (1971), 965-1004. MR 0322073 (48:437)
55.: -, Stability theorems for $\mathrm{K}_1$ , $\mathrm{K}_2$ and related functors modeled on Chevalley groups, Japan J. Math. (N.S.) 4 (1978), 77-108. MR 0528869 (81c:20031)
56.: A. V. Stepanov and N. A. Vavilov, Decomposition of transvections: a theme with variations, -Theory 19 (2000), 109-153. MR 1740757 (2000m:20076)
57.: K. Suzuki, Normality of the elementary subgroups of twisted Chevalley groups over commutative rings, J. Algebra 175 (1995), 526-536. MR 1339654 (96m:20077)
58.: G. Taddei, Normalité des groupes élémentaires dans les groupes de Chevalley sur un anneau, Applications of Algebraic -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)
59.: J. Tits, Systèmes générateurs de groupes de congruence, C. R. Acad. Sci. Paris Sér A-B 283 (1976), A693-A695. MR 0424966 (54:12924)
60.: L. N. Vaserstein, On the normal subgroups of $\GL_n$ over a ring, Algebraic -Theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980), Lecture Notes in Math., vol. 854, Springer, Berlin-New York, 1981, pp. 456-465. MR 0618316 (83c:20058)
61.: -, On normal subgroups of Chevalley groups over commutative rings, Tôhoku Math. J. (2) 38 (1986), 219-230. MR 0843808 (87k:20081)
62.: -, Normal subgroups of orthogonal groups over commutative rings, Amer. J. Math. 110 (1988), 955-973. MR 0961501 (89i:20071)
63.: -, Normal subgroups of symplectic groups over rings, -Theory 2 (1989), 647-673. MR 0999398 (90f:20064)
64.: L. N. Vaserstein and Hong You, Normal subgroups of classical groups over rings, J. Pure Appl. Algebra 105 (1995), 93-105. MR 1364152 (96k:20096)
65.: N. 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 (92k:20090)
66.: -, 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 (96c:20085)
67.: -, A third look at weight diagrams, Rend. Sem. Mat. Univ. Padova 104 (2000), 201-250. MR 1809357 (2001i:20099)
68.: -, Do it yourself structure constants for Lie algebras of types ${E}_l$ , Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 281 (2001), 60-104; English transl., J. Math. Sci. 120 (2004), no. 4, 1513-1548. MR 1875718 (2002k:17022)
69.: N. A. Vavilov and E. B. Plotkin, Chevalley groups over commutative rings. I. Elementary calculations, Acta Appl. Math. 45 (1996), 73-113. MR 1409655 (97h:20056)
70.: W. C. Waterhouse, Introduction to affine group schemes, Grad. Texts in Math., vol. 66, Springer-Verlag, New York-Berlin, 1979. MR 0547117 (82e:14003)
71.: J. S. Wilson, The normal and subnormal structure of general linear groups, Proc. Cambridge Philos. Soc. 71 (1972), 163-177. MR 0291304 (45:398)

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 20G15, 20G35

N. A. Vavilov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskii Prospekt 28, Staryi Peterhof, St. Petersburg 198904, Russia

M. R. Gavrilovich
Affiliation: Oxford University, Great Britain

DOI: 10.1090/S1061-0022-05-00871-X
PII: S 1061-0022(05)00871-X
Keywords: Chevalley groups, elementary subgroups, normal subgroups, standard description, minimal module, parabolic subgroups, decomposition of unipotents, root elements, orbit of the highest weight vector, the proof from the Book
Received by editor(s): 25/JUN/2003
Posted: June 21, 2005
Additional Notes: The present paper has been written in the framework of the RFBR projects 01-01-00924 (St. Petersburg State University), 03-01-00349 (POMI RAN). Part of the work was carried out during the authors' joint stay at the University of Bielefeld supported by SFB-343 and INTAS 00-566. At the final stage, the work 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, and the 2003 program of the Presidium of the Russian Academy of Sciences `Research in fundamental directions of modern mathematics'.
Copyright of article: Copyright 2005, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN: 1547-7371(e) ISSN: 1061-0022(p)

Previous issue \| Table of contents \| Next issue Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS