Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

Subring subgroups of symplectic groups in characteristic 2


Authors: A. Bak and A. Stepanov
Original publication: Algebra i Analiz, tom 28 (2016), nomer 4.
Journal: St. Petersburg Math. J. 28 (2017), 465-475
MSC (2010): Primary 14L15
DOI: https://doi.org/10.1090/spmj/1459
Published electronically: May 4, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In 2012, the second author obtained a description of the lattice of subgroups of a Chevalley group $ G(\Phi ,A)$ that contain the elementary subgroup $ E(\Phi ,K)$ over a subring $ K\subseteq A$ provided $ \Phi =B_n,$ $ C_n$, or $ F_4$, $ n\ge 2$, and $ 2$ is invertible in $ K$. It turned out that this lattice is a disjoint union of ``sandwiches'' parametrized by the subrings $ R$ such that $ K\subseteq R\subseteq A$. In the present paper, a similar result is proved in the case where $ \Phi =C_n$, $ n\ge 3$, and $ 2=0$ in $ K$. In this setting, more sandwiches are needed, namely those parametrized by the form rings $ (R,\Lambda )$ such that $ K\subseteq \Lambda \subseteq R\subseteq A$. The result generalizes Ya. N. Nuzhin's theorem of 2013 concerning the root systems $ \Phi =B_n,$ $ C_n$, $ n\ge 3$, where the same description of the subgroup lattice is obtained, but under the condition that $ A$ and $ K$ are fields such that $ A$ is algebraic over $ K$.


References [Enhancements On Off] (What's this?)

  • 1. E. Abe, Normal subgroups of Chevalley groups over commutative rings, Contemp. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1989, pp. 1-17. MR 991973
  • 2. E. Abe and K. Suzuki, On normal subgroups of Chevalley groups over commutative rings, Tohoku Math. J. (2) 28 (1976), no. 2, 185-198. MR 0439947
  • 3. A. Bak, The stable structure of quadratic modules, PhD. thesis, Columbia Univ., Columbia, USA, 1969.
  • 4. -, Odd dimension surgery groups of odd torsion groups vanish, Topology 14 (1975), no. 4, 367-374. MR 0400263
  • 5. -, K-theory of forms, Ann. of Math. Stud., vol. 98, Princeton Univ. Press, Princeton, NJ, 1981. MR 0632404
  • 6. -, Nonabelian K-theory: the nilpotent class of  $ \mathrm {K}_{1}$ and general stability, K-Theory 4 (1991), no. 4, 363-397. MR 1115826
  • 7. A. Bak and N. A. Vavilov, Normality for elementary subgroup functors, Math. Proc. Cambridge Philos. Soc. 118 (1995), no. 1, 35-47. MR 1329456
  • 8. -, Structure of hyperbolic unitary groups. I. Elementary subgroups, Algebra Colloq. 7 (2000), no. 2, 159-196. MR 1810843
  • 9. N. Bourbaki, Elements of mathematics. Lie groups and Lie algebras. Chapters 4-6, Springer-Verlag, Berlin, 2002. MR 1890629
  • 10. I. Z. Golubchik and A. V. Mikhalev, Generalized group identities in classical groups, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 114 (1982), 96-119; English transl., J. Soviet Math. 27 (1984), no. 4, 2904-2918. MR 669562
  • 11. N. L. Gordeev, Freedom in conjugacy classes of simple algebraic groups and identities with constants, Algebra i Analiz 9 (1998), no. 4, 63-78; English transl., St. Petersburg Math. J. 9 (1998), no. 4, 709-723. MR 1604024
  • 12. R. Hazrat, Dimension theory and nonstable $ K_1$ of quadratic modules, K-Theory 27 (2002), no. 4, 293-328. MR 1962906
  • 13. V. V. Nesterov and A. V. Stepanov, An identity with constans in a Chevalley group of type $ F_4$, Algebra i Analiz 21 (2009), no. 5, 196-202; English transl., St. Petersburg Math. J. 21 (2010), no. 5, 819-823. MR 2604568
  • 14. Ya. N. Nuzhin, Groups contained between groups of Lie type over different fields, Algebra i logika 22 (1983), no. 5, 526-541. (Russian) MR 759404
  • 15. -, Intermediate groups of Chevalley groups of type $ B_l$, $ C_l$, $ F_4$, $ G_2$ over nonperfect fields of characteristic 2 and 3, Sibirsk. Mat. Zh. 54 (2013), no. 1, 157-162; English transl., Sib. Math. J. 54 (2013), no. 1, 119-123. MR 3089335
  • 16. R. G. Steinberg, Lectures on Chevalley groups, Yale Univ., New Haven, Conn., 1968. MR 0466335
  • 17. A. V. Stepanov, Subring subgroups in Chevalley groups with doubly laced root systems, J. Algebra 362 (2012), 12-29. MR 2921626
  • 18. -, Elementary calculus in Chevalley groups over rings, J. Prime Res. Math. 9 (2013), 79-95. MR 3186522

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 14L15

Retrieve articles in all journals with MSC (2010): 14L15


Additional Information

A. Bak
Affiliation: Bielefeld University, Postfach 100131, 33501 Bielefeld, Germany
Email: bak@mathematik.uni-bielefeld.de

A. Stepanov
Affiliation: Mathematics and Mechanics Department, St. Petersburg State University, Universitetskiĭ pr. 28, 198504 St. Petersburg; St. Petersburg Electrotechnical University, Russia
Email: stepanov239@gmail.com

DOI: https://doi.org/10.1090/spmj/1459
Keywords: Symplectic group, commutative ring, subgroup lattice, Bak unitary group, group identity with constants, small unipotent element, nilpotent structure of $K1$
Received by editor(s): February 1, 2016
Published electronically: May 4, 2017
Additional Notes: The second author was supported by Russian Science Foundation, grant no. 14-11-00297
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society