Parametrization of generalized Heisenberg groups

  • Teerapong SuksumranEmail author
  • Sayan Panma
Original Paper


Let M be a left module over a ring R with identity and let \(\beta \) be a skew-symmetric R-bilinear form on M. The generalized Heisenberg group consists of the set \(M\times M\times R = \{(x, y, t):x, y\in M, t\in R\}\) with group law
$$\begin{aligned} (x_1, y_1, t_1)(x_2, y_2, t_2) = (x_1+x_2, y_1+y_2, t_1+\beta (x_1, y_2)+t_2). \end{aligned}$$
Under the assumption of 2 being a unit in R, we prove that the generalized Heisenberg group decomposes into a product of its subset and subgroup, similar to the well-known polar decomposition in linear algebra. This leads to a parametrization of the generalized Heisenberg group that resembles a parametrization of the Lorentz transformation group by relative velocities and space rotations.


Heisenberg group Parametrization of group Decomposition of group Polar decomposition Bilinear form 

Mathematics Subject Classification

20E22 20E34 15A63 16D10 



The authors would like to thank the referees and the editor for their careful reading of the manuscript and their useful comments. The authors were supported by Chiang Mai University.


  1. 1.
    Ferreira, M.: Hypercomplex Analysis and Applications. In: Sabadini, I., Sommen, F. (eds.) Chap. Gyrogroups in Projective Hyperbolic Clifford Analysis, Trends in Mathematics, pp. 61–80. Birkhäuser, Basel (2011)zbMATHGoogle Scholar
  2. 2.
    Foguel, T., Kinyon, M., Phillips, J.: On twisted subgroups and Bol loops of odd order. Rocky Mt. J. Math. 36, 183–212 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Folland, G.: Harmonic Analysis in Phase Space. Princeton University Press, Princeton (1989)CrossRefGoogle Scholar
  4. 4.
    Howe, R.: On the role of the Heisenberg group in harmonic analysis. Bull. Am. Math. Soc. (N.S.) 3(2), 821–843 (1980)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Sangkhanan, K., Suksumran, T.: On generalized Heisenberg groups: the symmetric case. Results. Math. 73, Article 91 (9 pp) (2018)Google Scholar
  6. 6.
    Suksumran, T.: Involutive groups, unique 2-divisibility, and related gyrogroup structures. J. Algebra Appl. 16(6), 1750114 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ungar, A.: Analytic Hyperbolic Geometry and Albert Einstein’s Special Theory of Relativity. World Scientific, Hackensack (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Research Center in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of ScienceChiang Mai UniversityChiang MaiThailand

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