Representation theory of SU(2)

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. It is the first case of a Lie group that is both a compact group and a non-abelian group. The first condition implies the representation theory is discrete: representations are direct sums of a collection of basic irreducible representations (governed by the Peter-Weyl theorem). The second means that irreducible representations will occur in dimensions greater than 1.

SU(2) is the universal covering group of SO(3), and so its representation theory includes that of the latter.

Lie algebra representations

Consider first representations of the Lie algebra

:$mathfrak\{su\}(2)$.

In principle this is the 'infinitesimal version' of SU(2); Lie algebras consist of infinitesimal transformations, and their Lie groups to 'integrated' transformations.

Then pass to the complex Lie algebra (i.e. complexify the Lie algebra). This doesn't affect the representation theory. The Lie algebra is spanned by three elements "e", "f" and "h" with the Lie brackets

:$[h,e]\; =e$:$[h,f]\; =-f$:$[e,f]\; =h$

Since $mathfrak\{su\}(2)$ is semisimple, the representation ρ(h) is always diagonalizable (for complex number scalars). Its eigenvalues are called the weights.

Suppose "x" is an eigenvector of the weight α. Then,

:$h\; [x]\; =alpha\; x$:$h\; [e\; [x]\; =(alpha\; +1)\; e\; [x]$:$h\; [f\; [x]\; =(alpha\; -1)\; f\; [x]$

In other words, "e" raises the weight by one and "f" reduces the weight by one. A consequence is that

:h²+ef+fe

is a Casimir invariant. By Schur's lemma, its action is proportional to the identity map, for irreducible representations. The constant of proportionality is conveniently written

:λ(λ+1).

Weights

A highest 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 representation is irreducible,

:$(h^2+ef+fe)\; x\; =\; (alpha^2\; +\; alpha)\; x=\; lambda\; (lambda\; +1)\; x$

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,

:$(alpha^2\; -\; alpha)\; x=lambda\; (lambda+1)\; x$

and so, α is either λ+1 or -λ.

Finite-dimensional representations only have finitely many weights, and so are both highest and lowest weight representations. For an irreducible finite-dimensional representation, the highest weight can't be less than the lowest weight. In addition, the difference between them has to be an integer because since

:$e\; [f\; [x]\; eq\; 0$

implies

:$f\; [x]\; eq\; 0$

and

:$f\; [e\; [x]\; eq\; 0$

implies

:$e\; [x]\; eq\; 0$.

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 highest 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 an integer or half-integer. 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).

Another approach

See under the example for Borel–Bott–Weil theorem.

ee also

* spin (physics)
* Rotation operator
* representation theory of SL2(R)