Advertisement

Journal of Mathematical Sciences

, Volume 243, Issue 4, pp 561–572 | Cite as

Products of Commutators on a General Linear Group Over a Division Algebra

  • E. A. EgorchenkovaEmail author
  • N. L. GordeevEmail author
Article
  • 5 Downloads

The word maps \( \tilde{w}:\kern0.5em {\mathrm{GL}}_m{(D)}^{2k}\to {\mathrm{GL}}_n(D) \) and \( \tilde{w}:\kern0.5em {D}^{\ast 2k}\to {D}^{\ast } \) for a word \( w=\prod \limits_{i=1}^k\left[{x}_i,{y}_i\right], \) where D is a division algebra over a field K, are considered. It is proved that if \( \tilde{w}\left({D}^{\ast 2k}\right)=\left[{D}^{\ast },{D}^{\ast}\right], \) then \( \tilde{w}\left({\mathrm{GL}}_n(D)\right)\supset {E}_n(D)\backslash Z\left({E}_n(D)\right), \) where En(D) is the subgroup of GLn(D), generated by transvections, and Z(En(D)) is its center. Furthermore if, in addition, n > 2, then \( \tilde{w}\left({E}_n(D)\right)\supset {E}_n(D)\backslash Z\left({E}_n(D)\right). \) The proof of the result is based on an analog of the “Gauss decomposition with prescribed semisimple part” (introduced and studied in two papers of the second author with collaborators) in the case of the group GLn(D), which is also considered in the present paper.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. Artin, Geometric Algebra [in Russian], Nauka, Moscow (1969).zbMATHGoogle Scholar
  2. 2.
    A. Borel, Linear Algebraic Groups, 2nd ed., Springer-Verlag, New York (1991).CrossRefGoogle Scholar
  3. 3.
    A. Borel, “On free subgroups of semisimple groups,” Enseign. Math., 29, 151–164 (1983); reproduced in: OEuvres - Collected Papers, vol. IV, Springer-Verlag, Berlin–Heidelberg (2001), pp. 41–54.Google Scholar
  4. 4.
    V. Chernousov, E. W. Ellers, and N. Gordeev, “Gauss decomposition with prescribed semisimple part: short proof,” J. Algebra, 229, No. 1, 314–332 (2000).MathSciNetCrossRefGoogle Scholar
  5. 5.
    A. Elkasapy and A. Thom, “About Gotô’s method showing surjectivity of word maps,” Indiana Univ. Math. J., 63, 1553–1565 (2014).MathSciNetCrossRefGoogle Scholar
  6. 6.
    E. W. Ellers and N. Gordeev, “On the conjectures of J. Thompson and O. Ore,” Trans. Amer. Math. Soc., 350, 3657–3671 (1998).MathSciNetCrossRefGoogle Scholar
  7. 7.
    E. W. Ellers and N. Gordeev, “Gauss decomposition with prescribed semisimple part in Chevalley groups. III. Finite twisted groups,” Commun. Algebra, 24, No. 14, 4447–4475 (1996).MathSciNetCrossRefGoogle Scholar
  8. 8.
    N. Gordeev, “Sums of orbits of algebraic groups. I,” J. Algebra, 295, No. 1, 62–80 (2006).MathSciNetCrossRefGoogle Scholar
  9. 9.
    N. L. Gordeev, B. È. Kunyavskii, and E. B. Plotkin, “Word maps and word maps with constants of simple algebraic groups,” Dokl. Math., 94, No. 3, 632–634 (2016).MathSciNetCrossRefGoogle Scholar
  10. 10.
    N. Gordeev, B. Kunyavskii, and E. Plotkin, “Word maps, word maps with constants and representation varieties of one-relator groups,” J. Algebra, 500, 390–424 (2018).MathSciNetCrossRefGoogle Scholar
  11. 11.
    N. Gordeev, B. Kunyavskii, and E. Plotkin, Word maps on perfect algebraic groups, arXiv:1801:00381 (2018).Google Scholar
  12. 12.
    N. Gordeev and J. Saxl, “Products of conjugacy classes in Chevalley groups over local rings,” St. Petersburg Math. J., 17, No. 2, 285–293 (2006).MathSciNetCrossRefGoogle Scholar
  13. 13.
    A. J. Hahn and O. T. O’Meara, The Classical Groups and K-theory, Springer-Verlag, Berlin–Heidelberg (1989).CrossRefGoogle Scholar
  14. 14.
    C. Y. Hui, M. Larsen, and A. Shalev, “TheWaring problem for Lie groups and Chevalley groups,” Israel J. Math., 210, 81–100 (2015).MathSciNetCrossRefGoogle Scholar
  15. 15.
    J. Morita and E. Plotkin, “Prescribed Gauss decompositions for Kac-Moody groups over fields,” Rend. Sem. Mat. Univ. Padova, 106, 153–163 (2001).MathSciNetzbMATHGoogle Scholar
  16. 16.
    V. P. Platonov and A. S. Rapinchuk, Algebraic Groups and Number Theory [in Russian], Nauka, Moscow (1991).zbMATHGoogle Scholar
  17. 17.
    T. A. Springer, Linear Algberaic Groups, 2nd ed., Birkhäuser, Boston–Basel–Berlin (1998).CrossRefGoogle Scholar
  18. 18.
    Sheng-Kui Ye, Sheng Chen, and Chun-Sheng Wang, “Gauss decomposition with prescribed semisimple part in quadratic groups,” Commun. Algebra, 37, 3054–3063 (2009).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Russian State Pedagogical UniversitySt.PetersburgRussia

Personalised recommendations