On Generalized Heisenberg Groups: The Symmetric Case
© 2018, Springer International Publishing AG, part of Springer Nature. In the literature, the famous Heisenberg group is the group of matrices of the form (1xz01y001),where x, y, and z are real numbers. In the present article, we examine a generalized Heisenberg group, obtained from an R-module M en...
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| Main Authors: | , |
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| Format: | Journal |
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2018
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| Subjects: | |
| Online Access: | https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85048585975&origin=inward http://cmuir.cmu.ac.th/jspui/handle/6653943832/58794 |
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| Institution: | Chiang Mai University |
| Summary: | © 2018, Springer International Publishing AG, part of Springer Nature. In the literature, the famous Heisenberg group is the group of matrices of the form (1xz01y001),where x, y, and z are real numbers. In the present article, we examine a generalized Heisenberg group, obtained from an R-module M endowed with an R-bilinear form β, where R is a ring with identity. We show that the structure of the generalized Heisenberg group and its generating space are intertwined. In particular, we prove that if β is symmetric, then the corresponding Heisenberg group possesses an involutive decomposition into subgroups, which eventually becomes the semidirect product of groups. This leads to a better understanding of the algebraic structure of the generalized Heisenberg group as well as its extensions by subgroups. |
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