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Toward the Reverse Decomposition of Unipotents. II. The Relative Case

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Recently, Raimund Preusser displayed very short polynomial expressions of elementary generators in classical groups over an arbitrary commutative ring as products of conjugates of an arbitrary matrix and its inverse by absolute elementary matrices. In particular, this provides very short proofs for description of normal subgroups. In 2018, the author discussed various generalizations of these results to exceptional groups, specifically those of types E6 and E7. Here, a further variation of Preusser’s wonderful idea is presented. Namely, in the case of GL(n, R), n ≥ 4, similar expressions of elementary transvections as conjugates of g ∈ GL(n, R) and g−1 by relative elementary matrices x ∈ E(n, J) and then x ∈ E(n, R, J), for an ideal J ⊴ R, are obtained. Again, in particular, this allows to give very short proofs for the description of subgroups normalized by E(n, J) or E(n, R, J), and thus also of subnormal subgroups in GL(n, R).

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References

  1. A. Bak, “Subgroups of the general linear group normalized by relative elementary subgroup,” in: Algebraic K-theory, Part II, Springer, Berlin (1982), pp. 1–22.

  2. A. Bak, “Nonabelian K-theory: the nilpotent class of K1 and general stability,” K-Theory, 4, 363–397 (1991).

    Article  MathSciNet  Google Scholar 

  3. A. Bak and N. Vavilov, “Structure of hyperbolic unitary groups. I. Elementary subgroups,” Algebra Colloq., 7, No. 2, 159–196 (2000).

    Article  MathSciNet  Google Scholar 

  4. H. Bass, “K-theory and stable algebra,” Inst. Hautes Études Sci. Publ. Math., No. 22, 5–60 (1964).

  5. H. Bass, “Unitary algebraic K-theory,” Lecture Notes Math., 343, 57–265 (1973).

    Article  MathSciNet  Google Scholar 

  6. Z. I. Borewicz and N. A. Vavilov, “Subgroups of the general linear group over a semilocal ring, containing the group of diagonal matrices,” Proc. Steklov. Inst. Math., 148, 41–54 (1980).

    MATH  Google Scholar 

  7. Z. I. Borewicz and N. A. Vavilov, “The distribution of subgroups in the general linear group over a commutative ring,” Proc. Steklov. Inst. Math., 183, 27–46 (1985).

    Google Scholar 

  8. J. L. Brenner, “The linear homogeneous group. III,” Ann. of Math., 71, 210–223 (1960).

    Article  MathSciNet  Google Scholar 

  9. G. Habdank, “Mixed commutator groups in classical groups and a classification of subgroups of classical groups normalized by relative elementary groups,” Dissertation, Univ. Bielefeld (1987).

  10. G. Habdank, “A classification of subgroups of Λ-quadratic groups normalized by relative elementary groups,” Adv. Math., 110, 191–233 (1995).

    Article  MathSciNet  Google Scholar 

  11. A. J. Hahn and O. T. O’Meara, The Classical Groups and K-theory, Springer, Berlin (1989).

  12. R. Hazrat and N. Vavilov, “Bak’s work on the K-theory of rings,” J. K-Theory, 4, No. 1, 1–65 (2009).

    Article  MathSciNet  Google Scholar 

  13. R. Hazrat, N. Vavilov, and Z. Zhang, “The commutators of classical groups,” J. Math. Sci., 222, No. 4, 466–515 (2017).

    Article  MathSciNet  Google Scholar 

  14. R. Hazrat and Z. Zhang, “Generalized commutator formula,” Commun. Algebra, 39, No. 4, 1441–1454 (2011).

    Article  MathSciNet  Google Scholar 

  15. Li Fuan and Liu Mulan, “Generalized sandwich theorem,” K-Theory, 1, 171–184 (1987).

    Article  MathSciNet  Google Scholar 

  16. R. Preusser, “Structure of hyperbolic unitary groups II. Classification of E-normal subgroups,” Algebra Colloq., 24, No. 2, 195–232 (2017).

    Article  MathSciNet  Google Scholar 

  17. R. Preusser, “Sandwich classification for GLn(R), O2n(R) and U2n(R, Λ) revisited,” J. Group Theory, 21, No. 1, 21–44 (2018).

    Article  MathSciNet  Google Scholar 

  18. R. Preusser, “Sandwich classification for O2n+1(R) and U2n+1(R, Δ) revisited,” J. Group Theory, 21, No. 4, 539–571 (2018).

    Article  MathSciNet  Google Scholar 

  19. A. Stepanov, “Structure of Chevalley groups over rings via universal localization,” J. Algebra, 450, 522–548 (2016).

    Article  MathSciNet  Google Scholar 

  20. A. V. Stepanov, “A new look at the decomposition of unipotents and the normal structure of Chevalley groups,” St. Petersburg J. Math., 28, No. 3, 411–419 (2017).

    Article  MathSciNet  Google Scholar 

  21. A. Stepanov and N. Vavilov, “Decomposition of transvections: a theme with variations,” K-Theory, 19, No. 2, 109–153 (2000).

    Article  MathSciNet  Google Scholar 

  22. L. N. Vaserstein, “On the normal subgroups of the GLn of a ring,” in: Algebraic K-Theory, Evanston 1980, Springer, Berlin (1981), pp. 454–465.

  23. L. N. Vaserstein, “The subnormal structure of general linear groups,” Math. Proc. Cambridge Phil. Soc., 99, No. 3, 425–431 (1986).

    Article  MathSciNet  Google Scholar 

  24. L. N. Vaserstein, “The subnormal structure of general linear groups over rings,” Math. Proc. Cambridge Phil. Soc., 108, No. 2, 219–229 (1990).

    Article  MathSciNet  Google Scholar 

  25. L. Vaserstein and Hong You, “Normal subgroups of classical groups over rings,” J. Pure Appl. Algebra, 105, 93–106 (1995).

    Article  MathSciNet  Google Scholar 

  26. N. Vavilov, “Subnormal structure of general linear group,” Math. Proc. Cambridge Phil. Soc., 107, 103–106 (1990).

    Article  MathSciNet  Google Scholar 

  27. N. A. Vavilov, “Towards the reverse decomposition of unipotents,” Zap. Nauchn. Semin. POMI, 470, 21–37 (2018).

    Google Scholar 

  28. N. A. Vavilov, “Unrelativised standard commutator formula,” Zap. Nauch. Semin. POMI, 470, 38–49 (2018).

    Google Scholar 

  29. N. A. Vavilov and A. V. Stepanov, “Standard commutator formula,” Vestn. St. Petersburg State Univ., Ser. 1, 41, No. 1, 5–8 (2008).

  30. N. A. Vavilov and A. V. Stepanov, “Standard commutator formulae, revisited,” Vestn. St. Petersburg State Univ., Ser. 1, 43, No. 1, 12–17 (2010).

  31. J. S. Wilson, “The normal and subnormal structure of general linear groups,” Proc. Cambridge Phil. Soc., 71, 163–177 (1972).

    Article  MathSciNet  Google Scholar 

  32. H. You, “Subgroups of classical groups normalized by relative elementary groups,” J. Pure Appl. Algebra, 216, 1040–1051 (2012).

    Article  MathSciNet  Google Scholar 

  33. H. You and X. Zhou, “Subnormal structure of classical groups over Banach algebra,” J. Algebra, 357, 222–230 (2012).

    Article  MathSciNet  Google Scholar 

  34. Z. Zhang, “Lower K-theory of unitary groups,” Ph. D. Thesis, Univ. Belfast (2007).

  35. Z. Zhang, “Stable sandwich classification theorem for classical-like groups,” Proc. Cambridge Phil. Soc., 143, 607–619 (2007).

    Article  MathSciNet  Google Scholar 

  36. Z. Zhang, “Subnormal structure of non-stable unitary groups over rings,” J. Pure Appl. Algebra, 214, No. 5, 622–628 (2010).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    N. Vavilov

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  1. N. Vavilov
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Correspondence to N. Vavilov.

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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 484, 2019, pp. 5–22.

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Vavilov, N. Toward the Reverse Decomposition of Unipotents. II. The Relative Case. J Math Sci 252, 749–760 (2021). https://doi.org/10.1007/s10958-021-05195-8

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