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In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups.
We look at representations of the Lie algebra <math>\mathfrak{su}(2)<math> first. We can then go to SU(2) by solving PDEs.
Let's complexify the Lie algebra first. It won't affect the representation theory (at least, if we don't consider unitary representations). Then, the Lie algebra is spanned by three elements e, f and h with the Lie brackets
- <math>[h,e]=e<math>
- <math>[h,f]=-f<math>
- <math>[e,f]=h<math>
Recall that since <math>\mathfrak{su}(2)<math> is semisimple, the representation ρ(h) is always diagonalizable (for real representations, complexify it). Its eigenvalues are called the weights.
Suppose x is an eigenvector of the weight α. Then,
- <math>h[x]=\alpha x<math>
- <math>h[e[x]]=(\alpha +1) e[x]<math>
- <math>h[f[x]]=(\alpha -1) f[x]<math>
In other words, e raises the weight by one and f reduces the weight by one.
h2+ef+fe is a Casimir invariant. By Schur's lemma, it is proportional to the identity for irreducible representations. The constant of proportionality is called λ(λ+1).
A heighest weight representation is a representation with a weight α which is greater than all the other weights.
If x is an eigenvector of α, e[x]=0.
If the rep is irreducible,
- <math>(\alpha^2 + \alpha) x= \lambda (\lambda +1) x<math>
and so, since x is nonzero, α is either λ or -λ-1.
A lowest weight representation is a representation with a weight α which is lower than all the other weights.
If x is an eigenvector of α, f[x]=0.
If the rep is irreducible,
- <math>(\alpha^2 - \alpha) x=\lambda (\lambda+1) x<math>
and so, α is either λ+1 or -λ.
Obviously, finite dimensional representations only have finitely many weights, and so, are both heightest and lowest weight representations.
for an irreducible finite dimensional representation, the heighest weight can't be less than the lowest weight. In addition, the difference between them has to be an integer because since <math>e[f[x]]\neq 0<math> implies <math>f[x] \neq 0<math> and <math>f[e[x]] \neq 0<math> implies <math>e[x] \neq 0<math>, if the difference isn't an integer, there will always be a weight which is one more or one less than any given weight, contradicting the assumption of finite dimensionality.
Since λ<λ+1 and -λ-1<-λ, without any loss of generality, we can assume the heighest weight is λ (if it's -λ-1, just redefine a new λ' as -λ-1) and the lowest weight would then have to be -λ. This means λ has to be a nonnegative integer multiple of half. Every weight is a number between λ and -λ which differs from them by an integer and has multiplicity one. This can be seen by assuming otherwise. Then, we can define a proper subrepresentation generated by an eigenvector of λ and f applied to it any number of times, contradicting the assumption of irreducibility.
This construction also shows for any given nonnegative integer multiple of half λ, all finite dimensional irreps with λ as its highest weight are equivalent (just make an identification of a highest weight eigenvector of one with one of the other).
See also spin (physics). SU(2) is the universal covering group of SO(3).
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