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 Compact semi-simple groups 
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Joined: 23 Oct 2010, 14:40
Posts: 2
Post Compact semi-simple groups
Hello Folks,

I just joined the forum. BS math and BS physics back in 1968. Interrupted my physics education for about 40 years to earn a living. Received R to R as a gift from youngest daughter and have reached chapter 13 over the last few months.

I have read the following two-sentence snippet several times and find my train of thought derailed with each reading. The source is on page 274 in section 13.7.

Compact (see 12.6) semi-simple groups have the property that all their representations are completely reducible. It is sufficient to study irreducible representations of such a group, every representation being just a direct sum of these irreducible ones.

If all the group representations are completely reducible, where does one find the irreducible representations of such a group.

Am I missing something obvious?

Regards,
JDJ


23 Oct 2010, 16:03
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Joined: 07 Jun 2008, 08:21
Posts: 235
Post Re: Compact semi-simple groups
Quote:
Compact (see 12.6) semi-simple groups have the property that all their representations are completely reducible. It is sufficient to study irreducible representations of such a group, every representation being just a direct sum of these irreducible ones.

At the beginning of the paragraph that contains your quote, Penrose is talking about semi-simple groups (compact and non-compact). He then says that all representations of compact semi-simple groups are completely reducible. The next sentence then says "It is sufficient to study irreducible representations of such a group......".
Here "such a group" must refer to all semi-simple groups not just compact semi-simple groups. The sentence about compact semi-simple groups should be considered as an aside.
If when reading the sentence "It is sufficient to study irreducible representations of such a group......" you give emphasis to the word irreducible, as I have indicated by emboldening it, and Penrose by italicizing it, I think it then makes more sense.


Last edited by vasco on 24 Oct 2010, 13:14, edited 2 times in total.

24 Oct 2010, 10:04

Joined: 23 Oct 2010, 14:40
Posts: 2
Post Re: Compact semi-simple groups
Thanks for the prompt response. I am sure you are right. I had been looking through too narrow a mental window in my reading. I will be following this forum as I go along through the book.


24 Oct 2010, 12:37
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