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## Main Question or Discussion Point

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!

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!