Parametrization of generalized Heisenberg groups

• Main Authors: Teerapong Suksumran, Sayan Panma
• Format: Journal
• Published: 2020
• Subjects: Mathematics
• Online Access: https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85075338498&origin=inward; http://cmuir.cmu.ac.th/jspui/handle/6653943832/67914
• Institution: Chiang Mai University

© 2019, Springer-Verlag GmbH Germany, part of Springer Nature. Let M be a left module over a ring R with identity and let β be a skew-symmetric R-bilinear form on M. The generalized Heisenberg group consists of the set M× M× R= { (x, y, t) : x, y∈ M, t∈ R} with group law (x1,y1,t1)(x2,y2,t2)=(x1+x2,y1+y2,t1+β(x1,y2)+t2).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.