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Compact semisimple groups http://www.roadtoreality.info/viewtopic.php?f=22&t=1724 
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Author:  James D Jones [ 23 Oct 2010, 16:03 ] 
Post subject:  Compact semisimple 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 twosentence 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) semisimple 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 
Author:  vasco [ 24 Oct 2010, 10:04 ] 
Post subject:  Re: Compact semisimple groups 
Quote: Compact (see 12.6) semisimple 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 semisimple groups (compact and noncompact). He then says that all representations of compact semisimple 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 semisimple groups not just compact semisimple groups. The sentence about compact semisimple 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. 
Author:  James D Jones [ 24 Oct 2010, 12:37 ] 
Post subject:  Re: Compact semisimple 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. 
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