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.
References
E. Abe, “Automorphisms of Chevalley groups over commutative rings,” Algebra Analiz, 5, No. 2, 74–90 (1993).
H. Bass, Algebraic K-Theory [Russian translation], Mir, Moscow (1973).
A. Borel, “Properties and linear representations of Chevalley groups” [Russian translation],” in: Seminar on Algebraic Groups, Mir, Moscow (1973), pp. 9–59.
N. Bourbaki, Lie Groups and Algebras [Russian translation], Chaps IV–VI, Moscow (1972).
N. A. Vavilov, “Computations in exceptional groups,” Vestn. Samara Univ., 33, No. 7, 11–24 (2007).
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).
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).
N. A. Vavilov and A. Yu. Luzgarev, “The normalizer of the Chevalley group of type E6,” Algebra Analiz, 19, No. 5, 35–62 (2007).
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).
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).
N. A. Vavilov and A. V. Stepanov, “The standard commutation formula,” Vestn. S.-Peterburg Gos. Univ., Ser. 1, No. 1, 9–14 (2008).
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.
I. Z. Golubchik, “Lie type groups over PI-rings,” Fund. Prikl. Mat., 3, No. 2, 399–424 (1997).
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).
R. Steinberg, Lectures on Chevalley Groups [Russian translation], Mir, Moscow (1975).
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).
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).
E. Abe, “Chevalley groups over local rings,” Tôhoku Math. J. (2), 21, No. 3, 474–494 (1969).
E. Abe, “Chevalley groups over commutative rings,” in: Proceedings of the Conference on Radical Theory (Sendai, 1988), Uchida Rokakuho, Tokyo (1989), pp. 1–23.
E. Abe, “Normal subgroups of Chevalley groups over commutative rings,” Contemp. Math., 83, 1–17 (1989).
E. Abe, “Chevalley groups over commutative rings: normal subgroups and automorphisms rings,” Contemp. Math., 184, 13–23 (1995).
E. Abe and J. Hurley, “Centers of Chevalley groups over commutative rings,” Comm. Algebra, 16, No. 1, 57–74 (1998).
E. Abe and K. Suzuki, “On normal subgroups of Chevalley groups over commutative rings,” Tôhoku Math. J., 28, No. 1, 185–198 (1976).
H. Azad, M. Barry, and G. M. Seitz, “On the structure of parabolic subgroups,” Comm. Algebra, 18, 551–562 (1990).
A. Bak, “The stable structure of quadratic modules,” Thesis, Columbia Univ. (1969).
A. Bak, “Subgroups of the general linear group normalized by relative elementary groups,” Lecture Notes Math., 967, 1–22 (1982).
A. Bak, “Nonabelian K-theory: The nilpotent class of K1 and general stability,” K-Theory, 4, 363–397 (1991).
A. Bak, R. Hazrat, and N. Vavilov, “Localization-completion strikes again: relative K 1 is nilpotent by abelian,” K-Theory (2008).
A. Bak and N. Vavilov, “Normality for elementary subgroups functors,” Math. Proc. Cambridge Phil. Soc., 118, No. 1, 35–47 (1995).
A. Bak and N. Vavilov, “Structure of hyperbolic unitary groups. I. Elementary subgroups,” Algebra Colloquium, 7, No. 2, 159–196 (2000).
A. Bak, R. Hazrat, and N. Vavilov, “Structure of hyperbolic unitary groups. II. Normal subgroups” Algebra Colloquium (to appear).
H. Bass, “K-theory and stable algebra,” Inst. Hautes Études Sci. Publ. Math., No. 22, 5–60 (1964).
H. Bass, “Unitary algebraic K-theory,” Lecture Notes Math., 343, 57–265 (1973).
R. Carter, Simple groups of Lie Type, John Wiley, London et al. (1972).
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).
D. L. Costa and G. E. Keller, “Radix redux: normal subgroups of symplectic groups,” J. Reine Angew. Math., 427, No. 1, 51–105 (1991).
D. L. Costa and G. E. Keller, “On the normal subgroups of G2(A),” Trans. Amer. Math. Soc., 351, No. 12, 5051–5088 (1999).
M. Demazure and A. Grothendieck, “Schémas en groupes. I, II, III,” Lecture Notes Math., 151, 1–564 (1971); 152, 1–654; 153, 1–529.
A. J. Hahn and O. T. O’Meara, The Classical Groups and K-Theory, Springer-Verlag, Berlin et al. (1989).
R. Hazrat, “Dimension theory and non-stable K1 of quadratic module,” K-Theory, 27, 293–327 (2002).
R. Hazrat and N. Vavilov, “K1 of Chevalley groups are nilpotent,” J. Pure Appl. Algebra, 179, 99–116 (2003).
R. Hazrat and N. Vavilov, “Bak’s work on K-theory of rings” (to appear).
Li Fuan and Liu Mulan, “Generalized sandwich theorem,” K-Theory, 1, 171–184 (1987).
H. Matsumoto, “Sur les sous-groupes arithmétiques des groupes semi-simples déployés,” Ann. Sci. École Norm. Sup. (4), 2, 1–62 (1969).
V. M. Petechuk, “Stability structure of linear group over rings,” Mat. Studii, 16, No. 1, 13–24 (2001).
G. E. Roehrle, “On the structure of parabolic subgroups in algebraic groups,” J. Algebra, 157, No. 1, 80–115 (1993).
A. Stavrova, “Normal structure of maximal parabolic subgroups in Chevalley groups over rings,” Algebra Colloquium (to appear).
M. R. Stein, “Generators, relations, and coverings of Chevalley groups over commutative rings,” Amer. J. Math., 93, No. 4, 965–1004 (1971).
A. Stepanov and N. Vavilov, “Decomposition of transvections: a theme with variations,” K-Theory, 19, 109–153 (2000).
K. Suzuki, “On normal subgroups of twisted Chevalley groups over local rings,” Sci. Rep. Tokyo Kyoiku Daigaku, 13, 237–249 (1997).
K. Suzuki, “Normality of the elementary subgroups of twisted Chevalley groups over commutative rings,” J. Algebra, 175, No. 3, 526–536 (1995).
G. Taddei, “Normalité des groupes élémentaire dans les groupes de Chevalley sur un anneau,” Contemp. Math., 55, 693–710 (1986).
L. N. Vaserstein, “On the normal subgroups of the GLn of a ring,” Lecture Notes Math., 854, 454–465 (1981).
L. N. Vaserstein, “The subnormal structure of general linear groups,” Math. Proc. Cambridge Phil. Soc., 99, 425–431 (1986).
L. N. Vaserstein, “On normal subgroups of Chevalley groups over commutative rings,” Tôhoku Math. J. (2), 36, No. 5, 219–230 (1986).
L. N. Vaserstein, “The subnormal structure of general linear groups over rings,” Math. Proc. Cambridge Phil. Soc., 108, No. 2, 219–229 (1990).
L. N. Vaserstein and You Hong, “Normal subgroups of classical groups over rings,” J. Pure Appl. Algebra, 105, No. 1, 93–106 (1995).
N. Vavilov, “A note on the subnormal structure of general linear groups,” Math. Proc. Cambridge Phil. Soc., 107, No. 2, 193–196 (1990).
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.
N. Vavilov, “A third look at weight diagrams,” Rendiconti del Rend. Sem. Mat. Univ. Padova, 204, No. 1, 201–250 (2000).
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).
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).
N. A. Vavilov and E. B. Plotkin, “Chevalley groups over commutative rings. I. Elementary calculations,” Acta Appl. Math., 45, 73–115 (1996).
Zhang, Zuhong, “Stable sandwich classification theorem for classical-like groups,” Math. Proc. Cambridge Phil. Soc., 143, No. 3, 607–619 (2007).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 349, 2007, pp. 30–52.
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Vavilov, N.A., Stavrova, A.K. Basic reductions in the description of normal subgroups. J Math Sci 151, 2949–2960 (2008). https://doi.org/10.1007/s10958-008-9019-1
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DOI: https://doi.org/10.1007/s10958-008-9019-1