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Embeddings of almost homogeneous Heisenberg groups. (English) Zbl 0955.22009
In an unpublished manuscript the author studied almost homogeneous topological groups \(G\), i.e. \(G\) is supposed to be a locally compact connected group whose automorphism group \(\Gamma\) acts on \(G\) with exactly three orbits, and proved that each such group is nilpotent of class 2, a so-called Heisenberg group. Heisenberg groups are characterized by some symplectic map \(\beta : V \times V \to Z\), where \(V\) and \(Z\) are real vector spaces of finite dimension. In a recent paper of the author [Forum Math. 11, 659-672 (1999; Zbl 0928.22008)] the almost homogeneous species of the Heisenberg groups have been completely classified and characterized. The key result states that a Heisenberg group \(G_\beta\) is almost homogeneous if and only if there is a compact group \(\Phi\) that induces orthogonal transformation groups on both \(V\) and \(Z\) acting transitively on the respective unit spheres such that \((v^\varphi,w^\varphi)^\beta = ((v,w)^\beta)^\varphi\) holds for any \(v,w \in V\) and \(\varphi \in \Phi\). There are three infinite series and six exceptional almost homogeneous Heisenberg groups, the latter being associated with the compact groups \(\Phi = \text{SO}_3{\mathbb R}\), \(\text{SU}_3{\mathbb C}\), \(\text{G}_2\), \(\text{U}_2{\mathbb H}\), \(\text{SU}_4{\mathbb C}\), \(\text{Spin}_7{\mathbb R}\).
In the current paper under review the author investigates embeddings of almost homogeneous Heisenberg groups into each other. Except for five cases which involve two of the exceptional Heisenberg groups and one general case, the author comes up with the complete graph of embeddings between Heisenberg groups.

22D45 Automorphism groups of locally compact groups
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