Representation theory of SU(2)

In the study of the representation theory

Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. In essence, a representation makes an abstract algebraic object concrete by describing its elements by matrices and the algebraic...

  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

Compact group

In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...

 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).

In the study of the representation theory

Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. In essence, a representation makes an abstract algebraic object concrete by describing its elements by matrices and the algebraic...

  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

Compact group

In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...

 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

Universal covering group

In mathematics, a covering group of a topological group H is a covering space G of H such that G is a topological group and the covering map p : G → H is a continuous group homomorphism. The map p is called the covering homomorphism...

 of SO(3), and so its representation theory includes that of the latter.

Lie algebra representations

Consider first representations of the Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

.

In principle this is the 'infinitesimal version' of SU(2); Lie algebras consist of infinitesimal transformation

Infinitesimal transformation

In mathematics, an infinitesimal transformation is a limiting form of small transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space. This is conventionally represented by a 3×3 skew-symmetric matrix A...

s, and their Lie groups to 'integrated' transformations.

Then pass to the complex Lie algebra (i.e. complexify

Complexification

In mathematics, the complexification of a real vector space V is a vector space V^C over the complex number field obtained by formally extending scalar multiplication to include multiplication by complex numbers...

 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
Since is semisimple

Semisimple

In mathematics, the term semisimple is used in a number of related ways, within different subjects. The common theme is the idea of a decomposition into 'simple' parts, that fit together in the cleanest way ....

, the representation ρ(h) is always diagonalizable (for complex number scalars). Its eigenvalues are called the weights

Weight (representation theory)

In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F, or equivalently, a one dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group...

.

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

Casimir invariant

In mathematics, a Casimir invariant or Casimir operator is a distinguished element of the centre of the universal enveloping algebra of a Lie algebra...

. By Schur's lemma

Schur's lemma

In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that that if M and N are two finite-dimensional irreducible representations...

, 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,
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,
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
implies
and
implies
.

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

Half-integer

In mathematics, a half-integer is a number of the form,where is an integer. For example,are all half-integers. Note that a half of an integer is not always a half-integer: half of an even integer is an integer but not a 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).