On Generalized Heisenberg Groups: The Symmetric Case

In the literature, the famous Heisenberg group is the group of matrices of the form 1xz01y001,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{pmatrix} 1 &{}\quad x &{}\quad z\\ 0 &{}\quad 1 &{}\quad y\\ 0 &{}\quad 0 &{}\quad 1 \end{pmatrix}, \end{aligned}$$\end{document}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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}, 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} 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.