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Parametrization of generalized Heisenberg groups
- Source :
- Applicable Algebra in Engineering, Communication and Computing. 32:135-146
- Publication Year :
- 2019
- Publisher :
- Springer Science and Business Media LLC, 2019.
-
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.
- Subjects :
- Algebra and Number Theory
Applied Mathematics
Lorentz transformation
Polar decomposition
020206 networking & telecommunications
0102 computer and information sciences
02 engineering and technology
01 natural sciences
Combinatorics
symbols.namesake
010201 computation theory & mathematics
Linear algebra
0202 electrical engineering, electronic engineering, information engineering
symbols
Heisenberg group
Beta (velocity)
Mathematics
Subjects
Details
- ISSN :
- 14320622 and 09381279
- Volume :
- 32
- Database :
- OpenAIRE
- Journal :
- Applicable Algebra in Engineering, Communication and Computing
- Accession number :
- edsair.doi...........0027b01923e4022804108d8241c484a3