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1. A noncompact Lie group has no nontrivial finite dimensional unitary linear irreps.

2. The linear representations of a simply connected Lie group are fully reducible.

3. The unitary linear irreps of a simply connected Lie group can be put in bijective correspondence with (essentially) selfadjoint irreps of the corresponding Lie algebra.

4. The Theorem of Nelson. The only reference for a proof that i found is the original article by Nelson, but, unfortunately, it's not within my reach.

The relevance of this thread: these mathematical results are fundamental in understanding the concept of implementing space-time symmetries in the Hilbert space language of QM through the so-called "Wigner-Weyl method".

Daniel.