Optional: A torus is contained in a maximal torus

Any torus is contained in a maximal torus

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We have so far proved that any compact matrix group G contains a nontrivial torus. We will now show that:


Any nontrivial torus is contained in a maximal torus (so that, in particular, G contains a maximal torus).


Suppose, for a contradiction, that we have a torus T 1 not contained in a maximal torus. Then T 1 cannot itself be maximal (it's contained in itself), so it must be contained in a strictly bigger torus T 2 , and T 2 is also not allowed to be maximal. Therefore T 2 is contained in a strictly bigger non-maximal torus T 3 . Continuing in this manner, I can construct a sequence of nested tori T 1 T 2 T 3 each strictly included in the next.

We now pass to the level of Lie algebras: 𝔱 1 𝔱 2 𝔱 3 . These are all subspaces of 𝔤 , which is a finite-dimensional vector space. Therefore this sequence of Lie algebras must "stabilise" at some point, that is there exists an N such that 𝔱 N + k = 𝔱 N for all k 0 .

We saw earlier that a torus is exp of its Lie algebra, so T k = exp ( 𝔱 k ) for all k . But then T N + k = T N for all k 0 , and this is a contradiction because T N T N + 1 is supposed to be a strict inclusion.


Now we have found a maximal torus t : U ( 1 ) n G ( T = t ( U ( 1 ) n ) ), we will use it in the following way. Given a (smooth, complex) representation R : G G L ( V ) , we can restrict to get a representation R | T : T G L ( V ) of our maximal torus. Because T U ( 1 ) n , we deduce that V = W λ where W λ = { v V : R ( t ( e i θ 1 , , e i θ n ) ) v = e i ( λ 1 θ 1 + + λ n θ n ) v } . Here, λ = ( λ 1 , , λ n ) .


The bigger our torus, the larger n is, so the more integers λ i we have to split up our weight spaces.


There is a more sophisticated way to think about λ . We can think of λ as the linear function λ 1 θ 1 + + λ n θ n of the θ s. The θ s are coordinates on a certain vector space. This is not quite the Lie algebra of T , because the Lie algebra of T consists of n -tuples of imaginary numbers ( i θ 1 , , i θ n ) (just like how 𝔲 ( 1 ) = i 𝐑 ). Instead, the vector ( θ 1 , , θ n ) lives in i 𝔱 𝔱 𝐂 . This means that λ lives most naturally in ( i 𝔱 ) * (where * means the dual space of linear functions). We will discuss this more formally when we talk about the Killing form.