# Parametrization of generalized Heisenberg groups

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## Abstract

Let Under the assumption of 2 being a unit in

*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$$\begin{aligned} (x_1, y_1, t_1)(x_2, y_2, t_2) = (x_1+x_2, y_1+y_2, t_1+\beta (x_1, y_2)+t_2). \end{aligned}$$

*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.## Keywords

Heisenberg group Parametrization of group Decomposition of group Polar decomposition Bilinear form## Mathematics Subject Classification

20E22 20E34 15A63 16D10## Notes

### Acknowledgements

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.

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