Classification of semi-simple Lie groups
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From: metroplex021 on 2 Jul 2010 12:15 A while ago I heard the following two facts about semi-simple Lie groups (though I have a feeling they may have to be restricted to connected semi-simple Lie groups): 1. That semi-simple Lie groups are classified by their weight (and co- weight) and root (and co-root) lattices; 2. That all of these lattices can be deduced from the fundamental representations of the group. (So that if we have a complete set of representations we can go on and infer the group.) Can someone confirm for me that these are indeed the case, or suggest a reference where the above are stated? (I am a physics graduate but with little pure math knowledge, so the more approachable the better.) Thanks a lot!
From: Timothy Murphy on 3 Jul 2010 09:51
metroplex021 wrote: > A while ago I heard the following two facts about semi-simple Lie > groups (though I have a feeling they may have to be restricted to > connected semi-simple Lie groups): > > 1. That semi-simple Lie groups are classified by their weight (and co- > weight) and root (and co-root) lattices; > 2. That all of these lattices can be deduced from the fundamental > representations of the group. (So that if we have a complete set of > representations we can go on and infer the group.) > > Can someone confirm for me that these are indeed the case, or suggest > a reference where the above are stated? (I am a physics graduate but > with little pure math knowledge, so the more approachable the > better.) Thanks a lot! I think Lie groups are normally assumed to be connected. Adams, "Lectures on Lie Groups", is a very nice book on the subject. It deals almost entirely with semisimple Lie groups. (I don't think the associated book by Adams on Exceptional Lie Groups is nearly so good; I believe it was completed by a student of Adams after Adams' tragic early death.) -- Timothy Murphy e-mail: gayleard /at/ eircom.net tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland |