The Road to Realityhttp://www.roadtoreality.info/ Compact semi-simple groupshttp://www.roadtoreality.info/viewtopic.php?f=22&t=1724 Page 1 of 1

 Author: James D Jones [ 23 Oct 2010, 16:03 ] Post subject: 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

 Author: vasco [ 24 Oct 2010, 10:04 ] Post subject: 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.

 Author: James D Jones [ 24 Oct 2010, 12:37 ] Post subject: 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.

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