Group theory proofs

  • #1
dextercioby
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It's always annoying when one finds in books (written by (theoretical) physicists for (theoretical) physics students) statements like those below without a mere cross-reference for a mathematically-rigurous proof. And that's what i'm searching for right now: either point me to some books, or post the proofs right here (that would be perfect). So here's what's been bugging me lately:

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.
 

Answers and Replies

  • #2
matt grime
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The underlying idea is the following:

any lie group homomorphism is determined uniquely by the induced lie algebra map.

See Fulton and Harris, Representation Theory (A First Course) Springer GTM 129 (1991) Lecture 8 onwards.

I don't know that it'll answer all your questions, but it seems like it'll be a start in the right direction. It is quite a cheap and readily available book. You library should have a copy.
 

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