Abstract
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
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.
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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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This research is part of the project On generalized Heisenberg groups and their modules with Lie bracket, supported by the Thailand Research Fund and the Office of the Higher Education Commission, Thailand via the Research Grant for New Scholar (MRG), Year 2018, under Grant No. MRG6180032.
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Suksumran, T., Panma, S. Parametrization of generalized Heisenberg groups. AAECC 32, 135–146 (2021). https://doi.org/10.1007/s00200-019-00405-y
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DOI: https://doi.org/10.1007/s00200-019-00405-y