Parametrization of generalized Heisenberg groups

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

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

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.