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.