Journal of Lie Theory 27 (2017), No. 2, 569--578
Copyright Heldermann Verlag 2017
Compact Elements in Connected Lie Groups
Mikhail Kabenyuk
Institute of Fundamental Sciences, Kemerovo State University, 650043 Kemerovo, Russia
kabenyuk@kemsu.ru
We prove that the set of compact elements in the group extension of the 3-dimensional Heisenberg group by SO(2) (the so-called oscillator group) is not dense. We also give a new proof of the following criterion: The set of compact elements of a connected Lie group G is dense in G if and only if every Cartan subgroup of G is compact.
Keywords: Lie group, compact element, Heisenberg group, oscillator group, Cartan subgroup.
MSC: 22C05, 22E15, 22E25
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