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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

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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