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 there will be irreducible representations 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, by dint of a surjective homomorphism to it. This underlies the significance of SU(2) for the description of non-relativistic spin in theoretical physics; see below for other physical and historical context.
As shown below, the finite-dimensional irreducible representations of SU(2) are indexed by a non-negative integer formula_1 and have dimension formula_2. In the physics literature, the representations are labeled by the quantity formula_3, where formula_4 is then either an integer or a half-integer, and the dimension is formula_5.
Lie algebra representations.
The representations of the group are found by considering representations of su(2), the Lie algebra of SU(2). Since the group SU(2) is simply connected, every representation of its Lie algebra can be integrated to a group representation; we will give an explicit construction of the representations at the group level below. A reference for this material is Section 4.6 of .
Real and complexified Lie algebras.
The real Lie algebra su(2) has a basis given by
(These basis matrices are related to the Pauli matrices by formula_7 and formula_8) 
The matrices are a representation of the quaternions:
where is the conventional 2×2 identity matrix:formula_12
Consequently, the commutator brackets of the matrices satisfy
It is then convenient to pass to the complexified Lie algebra
(Skew self-adjoint matrices with trace zero plus self-adjoint matrices with trace zero gives all matrices with trace zero.) As long as we are working with representations over formula_15 this passage from real to complexified Lie algebra is harmless. The reason for passing to the complexification is that it allows us to construct a nice basis of a type that does not exist in the real Lie algebra su(2).
The complexified Lie algebra is spanned by three elements formula_16, formula_17, and formula_18, given by
or, explicitly, 
These satisfy the commutation relations
Up to a factor of 2, the elements formula_18, formula_16 and formula_17 may be identified with the angular momentum operators formula_25, formula_26, and formula_27, respectively. The factor of 2 is a discrepancy between conventions in math and physics; we will attempt to mention both conventions in the results that follow.
Weights and the structure of the representation.
In this setting, the eigenvalues for formula_18 are referred to as the weights of the representation. The following elementary result is a key step in the analysis. Suppose that formula_29 is an eigenvector for formula_18 with eigenvalue formula_31, that is, that formula_32. Then
In other words, formula_34 is either the zero vector or an eigenvector for formula_18 with eigenvalue formula_36 and formula_37 is either zero or an eigenvector for formula_18 with eigenvalue formula_39. Thus, the operator formula_16 acts as a raising operator, increasing the weight by 2, while formula_17 acts as a lowering operator. 
Suppose now that formula_42 is an irreducible, finite-dimensional representation of the complexified Lie algebra. Then formula_18 can have only finitely many eigenvalues. In particular, there must be an eigenvalue formula_44 with the property that formula_45 is not an eigenvalue. Let formula_46 be an eigenvector for formula_18 with eigenvalue formula_48:
Then we must have
or else the above identity would tell us that formula_51 is an eigenvector with eigenvalue formula_52.
Now define a "chain" of vectors formula_53 by
A simple argument by induction then shows that
for all formula_56. Now, if formula_57 is not the zero vector, it is an eigenvector for formula_18 with eigenvalue formula_59. Since, again, formula_18 has only finitely many eigenvectors, we conclude that formula_61 must be zero for some formula_4 (and then formula_63 for all formula_64). 
Let formula_65 be the last nonzero vector in the chain; that is, formula_66 but formula_67. Then of course formula_68 and by the above identity with formula_69, we have
Since formula_2 is at least one and formula_66, we conclude that formula_48 "must be equal to the non-negative integer" formula_1. 
We thus obtain a chain of formula_2 vectors formula_76 such that formula_17 acts as
and formula_16 acts as
and formula_18 acts as
(We have replaced formula_48 with its currently known value of formula_1 in the above formulas.) 
Since the vectors formula_57 are eigenvectors for formula_18 with distinct eigenvalues, they must be linearly independent. Furthermore, the span of formula_87 is clearly invariant under the action of the complexified Lie algebra. Since formula_42 is assumed irreducible, this span must be all of formula_42. We thus obtain a complete description of what an irreducible representation must look like; that is, a basis for the space and a complete description of how the generators of the Lie algebra act. Conversely, for any formula_90 we can construct a representation by simply using the above formulas and checking that the commutation relations hold. This representation can then be shown to be irreducible.
Conclusion: For each non-negative integer formula_1, there is a unique irreducible representation with highest weight formula_1. Each irreducible representation is equivalent to one of these. The representation with highest weight formula_1 has dimension formula_2 with weights formula_95, each having multiplicity one.
The Casimir element.
We now introduce the (quadratic) Casimir element, formula_96 given by
We can view formula_96 as an element of the universal enveloping algebra or as an operator in each irreducible representation. Viewing formula_96 as an operator on the representation with highest weight formula_1, we may easily compute that formula_96 commutes with each formula_102. Thus, by Schur's lemma, formula_96 acts as a scalar multiple formula_104 of the identity for each formula_1. We can write formula_96 in terms of the formula_107 basis as follows:
which simplifies to
The eigenvalue of formula_96 in the representation with highest weight formula_1 can be computed by applying formula_96 to the highest weight vector, which is annihilated by formula_16. Thus, we get 
In the physics literature, the Casimir is normalized as formula_115. Labeling things in terms of formula_3, the eigenvalue formula_117 of formula_118 is then computed as
The group representations.
Action on polynomials.
Since SU(2) is simply connected, a general result shows that every representation of its (complexified) Lie algebra gives rise to a representation of SU(2) itself. It is desirable, however, to give an explicit realization of the representations at the group level. The group representations can be realized on spaces of polynomials in two complex variables. That is, for each non-negative integer formula_1, we let formula_121 denote the space of homogeneous polynomials of degree formula_1 in two complex variables. Then the dimension of formula_121 is formula_2. There is a natural action of SU(2) on each formula_121, given by
The associated Lie algebra representation is simply the one described in the previous section. (See here for an explicit formula for the action of the Lie algebra on the space of polynomials.)
The characters.
The character of a representation formula_127 is the function formula_128 given by
Characters plays an important role in the representation theory of compact groups. The character is easily seen to be a class function, that is, invariant under conjugation.
In the SU(2) case, the fact that the character is a class function means it is determined by its value on the maximal torus formula_130 consisting of the diagonal matrices in SU(2). Since the irreducible representation with highest weight formula_1 has weights formula_95, it is easy to see that the associated character satisfies
This expression is a finite geometric series that can be simplified to
This last expression is just the statement of the Weyl character formula for the SU(2) case.
Actually, following Weyl's original analysis of the representation theory of compact groups, one can classify the representations entirely from the group perspective, without using Lie algebra representations at all. In this approach, the Weyl character formula plays an essential part in the classification, along with the Peter–Weyl theorem. The SU(2) case of this story is described here.
Relation to the representations of SO(3).
Note that either all of the weights of the representation are even (if formula_1 is even) or all of the weights are odd (if formula_1 is odd). In physical terms, this distinction is important: The representations with even weights correspond to ordinary representations of the rotation group SO(3). By contrast, the representations with odd weights correspond to double-valued (spinorial) representation of SO(3), also known as projective representations. 
In the physics conventions, formula_1 being even corresponds to formula_4 being an integer while formula_1 being odd corresponds to formula_4 being a half-integer. These two cases are described as integer spin and half-integer spin, respectively. The representations with odd, positive values of formula_1 are faithful representations of SU(2), while the representations of SU(2) with non-negative, even formula_1 are not faithful.
Another approach.
See under the example for Borel–Weil–Bott theorem.
Most important irreducible representations and their applications.
Representations of SU(2) describe non-relativistic spin, due to being a double covering of the rotation group of Euclidean 3-space. Relativistic spin is described by the representation theory of SL2(C), a supergroup of SU(2), which in a similar way covers SO+(1;3), the relativistic version of the rotation group. SU(2) symmetry also supports concepts of isobaric spin and weak isospin, collectively known as "isospin".
The representation with formula_143 (i.e., formula_144 in the physics convention) is the 2 representation, the fundamental representation of SU(2). When an element of SU(2) is written as a complex matrix, it is simply a multiplication of column 2-vectors. It is known in physics as the spin-½ and, historically, as the multiplication of quaternions (more precisely, multiplication by a unit quaternion). This representation can also be viewed as a double-valued projective representation of the rotation group SO(3).
The representation with formula_145 (i.e., formula_146) is the 3 representation, the adjoint representation. It describes 3-d rotations, the standard representation of SO(3), so real numbers are sufficient for it. Physicists use it for the description of massive spin-1 particles, such as vector mesons, but its importance for spin theory is much higher because it anchors spin states to the geometry of the physical 3-space. This representation emerged simultaneously with the 2 when William Rowan Hamilton introduced versors, his term for elements of SU(2). Note that Hamilton did not use standard group theory terminology since his work preceded Lie group developments.
The formula_147 (i.e. formula_148) representation is used in particle physics for certain baryons, such as the Δ.