Groups normalized by the odd unitary group

Voronetsky, E.

doi:10.1090/spmj/1630

Groups normalized by the odd unitary group
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by E. Voronetsky
Translated by: E. Voronetsky

St. Petersburg Math. J. 31 (2020), 939-967

DOI: https://doi.org/10.1090/spmj/1630

Published electronically: October 27, 2020

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Abstract:

Quadratic forms on bimodules are defined and the sandwich classification theorem is proved for subgroups of the general linear group $\operatorname {GL}(P)$ normalized by the elementary unitary group $\operatorname {EU}(P)$ if $P$ is a regular bimodule with sufficiently large hyperbolic part.

References

E. Artin, Geometric algebra, Interscience Publishers, Inc., New York-London, 1957. MR 0082463
N. A. Vavilov and V. A. Petrov, On supergroups of $\textrm {EO}(2l,R)$, 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, DOI 10.1023/A:1023442407926
N. A. Vavilov and V. A. Petrov, On overgroups of $\textrm {Ep}(2l,R)$, Algebra i Analiz 15 (2003), no. 4, 72–114 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 15 (2004), no. 4, 515–543. MR 2068980, DOI 10.1090/S1061-0022-04-00820-9
N. A. Vavilov and V. A. Petrov, On overgroups of $\textrm {EO}(n,R)$, Algebra i Analiz 19 (2007), no. 2, 10–51 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 19 (2008), no. 2, 167–195. MR 2333895, DOI 10.1090/S1061-0022-08-00992-8
N. A. Vavilov and A. V. Stepanov, Overgroups of semisimple groups, Vestn. Samar. Gos. Univ. Estestvennonauchn. Ser. 3 (2008), 51–95 (Russian, with English and Russian summaries). MR 2473730
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
L. N. Vaseršteĭn, Stabilization for the classical groups over rings, Mat. Sb. (N.S.) 93(135) (1974), 268–295, 327 (Russian). MR 0338208
I. Z. Golubchik, Subgroups of the general linear group $\textrm {GL}_{n}(R)$ over an associative ring $R$, Uspekhi Mat. Nauk 39 (1984), no. 1(235), 125–126 (Russian). MR 733962
V. A. Petrov, Odd unitary groups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 305 (2003), no. Vopr. Teor. Predst. Algebr. i Grupp. 10, 195–225, 241 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 130 (2005), no. 3, 4752–4766. MR 2033642, DOI 10.1007/s10958-005-0372-z
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
A. Bak, The stable structure of quadratic modules, Thesis, Columbia Univ., 1969.
Anthony Bak, $K$-theory of forms, Annals of Mathematics Studies, No. 98, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1981. MR 632404
Anthony Bak, Nonabelian $K$-theory: the nilpotent class of $K_1$ and general stability, $K$-Theory 4 (1991), no. 4, 363–397. MR 1115826, DOI 10.1007/BF00533991
A. Bak, R. Hazrat, and N. Vavilov, Localization-completion strikes again: relative $K_1$ is nilpotent by abelian, J. Pure Appl. Algebra 213 (2009), no. 6, 1075–1085. MR 2498798, DOI 10.1016/j.jpaa.2008.11.014
Anthony Bak and Raimund Preusser, The E-normal structure of odd dimensional unitary groups, J. Pure Appl. Algebra 222 (2018), no. 9, 2823–2880. MR 3783021, DOI 10.1016/j.jpaa.2017.11.002
Anthony Bak and Nikolai Vavilov, Normality for elementary subgroup functors, Math. Proc. Cambridge Philos. Soc. 118 (1995), no. 1, 35–47. MR 1329456, DOI 10.1017/S0305004100073436
Anthony Bak and Nikolai Vavilov, Structure of hyperbolic unitary groups. I. Elementary subgroups, Algebra Colloq. 7 (2000), no. 2, 159–196. MR 1810843, DOI 10.1007/s100110050017
H. Bass, $K$-theory and stable algebra, Inst. Hautes Études Sci. Publ. Math. 22 (1964), 5–60. MR 174604
Hyman Bass, Unitary algebraic $K$-theory, Algebraic $K$-theory, III: Hermitian $K$-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
Jean Dieudonné, On the automorphisms of the classical groups. With a supplement by Loo-Keng Hua, Mem. Amer. Math. Soc. 2 (1951), vi+122. MR 45125
Jean Dieudonné, Sur les groupes classiques, Publications de l’Institut de Mathématique de l’Université de Strasbourg, VI, Hermann, Paris, 1973 (French). Troisième édition revue et corrigée. MR 0344355
Jean A. Dieudonné, La géométrie des groupes classiques, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 5, Springer-Verlag, Berlin-New York, 1971 (French). Troisième édition. MR 0310083
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, DOI 10.1007/978-3-662-13152-7
Roozbeh Hazrat, Dimension theory and nonstable $K_1$ of quadratic modules, $K$-Theory 27 (2002), no. 4, 293–328. MR 1962906, DOI 10.1023/A:1022623004336
Roozbeh Hazrat and Nikolai Vavilov, Bak’s work on the $K$-theory of rings, J. K-Theory 4 (2009), no. 1, 1–65. MR 2538715, DOI 10.1017/is008008012jkt087
Hong You, Overgroups of symplectic group in linear group over commutative rings, J. Algebra 282 (2004), no. 1, 23–32. MR 2095570, DOI 10.1016/j.jalgebra.2004.07.036
Hong You, Overgroups of classical groups in linear group over Banach algebras, J. Algebra 304 (2006), no. 2, 1004–1013. MR 2264287, DOI 10.1016/j.jalgebra.2006.03.024
Max-Albert Knus, Quadratic and Hermitian forms over rings, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 294, Springer-Verlag, Berlin, 1991. With a foreword by I. Bertuccioni. MR 1096299, DOI 10.1007/978-3-642-75401-2
Fu An Li, The structure of symplectic groups over arbitrary commutative rings, Acta Math. Sinica (N.S.) 3 (1987), no. 3, 247–255. MR 916269, DOI 10.1007/BF02560038
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
Viktor Petrov, Overgroups of unitary groups, $K$-Theory 29 (2003), no. 3, 147–174. MR 2028500, DOI 10.1023/B:KTHE.0000006934.95243.91
Raimund Preusser, Structure of hyperbolic unitary groups II: Classification of E-normal subgroups, Algebra Colloq. 24 (2017), no. 2, 195–232. MR 3639030, DOI 10.1142/S1005386717000128
Raimund Preusser, Sandwich classification for $\textrm {GL}_n(R)$, $\textrm {O}_{2n}(R)$ and $\textrm {U}_{2n}(R,\Lambda )$ revisited, J. Group Theory 21 (2018), no. 1, 21–44. MR 3739342, DOI 10.1515/jgth-2017-0028
Raimund Preusser, Sandwich classification for $O _{2n+1}(R)$ and $U_{2n+1}(R,\Delta )$ revisited, J. Group Theory 21 (2018), no. 4, 539–571. MR 3819539, DOI 10.1515/jgth-2018-0011
Guoping Tang, Hermitian groups and $K$-theory, $K$-Theory 13 (1998), no. 3, 209–267. MR 1609905, DOI 10.1023/A:1007725531627
J. Tits, Formes quadratiques, groupes orthogonaux et algèbres de Clifford, Invent. Math. 5 (1968), 19–41 (French). MR 230747, DOI 10.1007/BF01404536
L. N. Vaserstein, Normal subgroups of orthogonal groups over commutative rings, Amer. J. Math. 110 (1988), no. 5, 955–973. MR 961501, DOI 10.2307/2374699
L. N. Vaserstein, Normal subgroups of symplectic groups over rings, Proceedings of Research Symposium on $K$-Theory and its Applications (Ibadan, 1987), 1989, pp. 647–673. MR 999398, DOI 10.1007/BF00535050
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, DOI 10.1016/0022-4049(94)00144-8
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, DOI 10.1017/CBO9780511565823.018
C. T. C. Wall, On the axiomatic foundations of the theory of Hermitian forms, Proc. Cambridge Philos. Soc. 67 (1970), 243–250. MR 251054, DOI 10.1017/s0305004100045515
André Weil, Algebras with involutions and the classical groups, J. Indian Math. Soc. (N.S.) 24 (1960), 589–623 (1961). MR 136682
J. S. Wilson, The normal and subnormal structure of general linear groups, Proc. Cambridge Philos. Soc. 71 (1972), 163–177. MR 291304, DOI 10.1017/s0305004100050416