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
m
{\displaystyle m}
and have dimension
m
+
1
{\displaystyle m+1}
. In the physics literature, the representations
are labeled by the quantity
l
=
m
/
2
{\displaystyle l=m/2}
, where
l
{\displaystyle l}
is then either an integer or a half-integer,
and the dimension is
2
l
+
1
{\displaystyle 2l+1}
.
== 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 (Hall 2015).
=== Real and complexified Lie algebras ===
The real Lie algebra su(2) has a basis given
by
u
1
=
(
0
i
i
0
)
u
2
=
(
0
−
1
1
0
)
u
3
=
(
i
0
0
−
i
)
,
{\displaystyle u_{1}={\begin{pmatrix}0&i\\i&0\end{pmatrix}}\qquad
u_{2}={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}\qquad
u_{3}={\begin{pmatrix}i&0\\0&-i\end{pmatrix}}~,}
which satisfy
[
u
1
,
u
2
]
=
2
u
3
;
[
u
2
,
u
3
]
=
2
u
1
;
[
u
3
,
u
1
]
=
2
u
2
.
{\displaystyle [u_{1},u_{2}]=2u_{3};\quad
[u_{2},u_{3}]=2u_{1};\quad [u_{3},u_{1}]=2u_{2}.}
It is then convenient to pass to the complexified
Lie algebra
s
u
(
2
)
+
i
s
u
(
2
)
=
s
l
(
2
;
C
)
{\displaystyle \mathrm {su} (2)+i\mathrm {su}
(2)=\mathrm {sl} (2;\mathbb {C} )}
.(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
C
{\displaystyle \mathbb {C} }
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
X
{\displaystyle X}
,
Y
{\displaystyle Y}
, and
H
{\displaystyle H}
, given by
H
=
1
i
u
3
;
X
=
1
2
i
(
u
1
−
i
u
2
)
;
Y
=
1
2
i
(
u
1
+
i
u
2
)
,
{\displaystyle H={\frac {1}{i}}u_{3};\quad
X={\frac {1}{2i}}(u_{1}-iu_{2});\quad Y={\frac
{1}{2i}}(u_{1}+iu_{2}),}
or, explicitly,
H
=
(
1
0
0
−
1
)
;
X
=
(
0
1
0
0
)
;
Y
=
(
0
0
1
0
)
.
{\displaystyle H={\begin{pmatrix}1&0\\0&-1\end{pmatrix}};\quad
X={\begin{pmatrix}0&1\\0&0\end{pmatrix}};\quad
Y={\begin{pmatrix}0&0\\1&0\end{pmatrix}}~.}
These satisfy the commutation relations
[
H
,
X
]
=
2
X
,
[
H
,
Y
]
=
−
2
Y
,
[
X
,
Y
]
=
H
{\displaystyle [H,X]=2X,\quad [H,Y]=-2Y,\quad
[X,Y]=H}
.Up to a factor of 2, the elements
H
{\displaystyle H}
,
X
{\displaystyle X}
and
Y
{\displaystyle Y}
may be identified with the angular momentum
operators
J
z
{\displaystyle J_{z}}
,
J
+
{\displaystyle J_{+}}
, and
J
−
{\displaystyle J_{-}}
, 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
H
{\displaystyle H}
are referred to as the weights of the representation.
The following elementary result is a key step
in the analysis. Suppose that
v
{\displaystyle v}
is an eigenvector for
H
{\displaystyle H}
with eigenvalue
α
{\displaystyle \alpha }
, that is, that
H
⋅
v
=
α
v
{\displaystyle H\cdot v=\alpha v}
. Then
H
⋅
(
X
⋅
v
)
=
(
α
+
2
)
X
⋅
v
H
⋅
(
Y
⋅
v
)
=
(
α
−
2
)
Y
⋅
v
{\displaystyle {\begin{aligned}H\cdot (X\cdot
v)&=(\alpha +2)X\cdot v\\[3pt]H\cdot (Y\cdot
v)&=(\alpha -2)Y\cdot v\end{aligned}}}
In other words,
X
⋅
v
{\displaystyle X\cdot v}
is either the zero vector or an eigenvector
for
H
{\displaystyle H}
with eigenvalue
α
+
2
{\displaystyle \alpha +2}
and
Y
⋅
v
{\displaystyle Y\cdot v}
is either zero or an eigenvector for
H
{\displaystyle H}
with eigenvalue
α
−
2
{\displaystyle \alpha -2}
. Thus, the operator
X
{\displaystyle X}
acts as a raising operator, increasing the
weight by 2, while
Y
{\displaystyle Y}
acts as a lowering operator.
Suppose now that
V
{\displaystyle V}
is an irreducible, finite-dimensional representation
of the complexified Lie algebra. Then
H
{\displaystyle H}
can have only finitely many eigenvalues. In
particular, there must be an eigenvalue
λ
∈
C
{\displaystyle \lambda \in \mathbb {C} }
with the property that
λ
+
2
{\displaystyle \lambda +2}
is not an eigenvalue. Let
v
0
{\displaystyle v_{0}}
be an eigenvector for
H
{\displaystyle H}
with eigenvalue
λ
{\displaystyle \lambda }
:
H
⋅
v
0
=
λ
v
0
{\displaystyle H\cdot v_{0}=\lambda v_{0}}
.Then we must have
X
⋅
v
0
=
0
{\displaystyle X\cdot v_{0}=0}
,or else the above identity would tell us
that
X
⋅
v
0
{\displaystyle X\cdot v_{0}}
is an eigenvector with eigenvalue
λ
+
2
{\displaystyle \lambda +2}
.
Now define a "chain" of vectors
v
0
,
v
1
,
…
{\displaystyle v_{0},v_{1},\ldots }
by
v
k
=
Y
k
⋅
v
0
{\displaystyle v_{k}=Y^{k}\cdot v_{0}}
.A simple argument by induction then shows
that
X
⋅
v
k
=
k
(
λ
−
(
k
−
1
)
)
v
k
−
1
{\displaystyle X\cdot v_{k}=k(\lambda -(k-1))v_{k-1}}
for all
k
=
1
,
2
,
…
{\displaystyle k=1,2,\ldots }
. Now, if
v
k
{\displaystyle v_{k}}
is not the zero vector, it is an eigenvector
for
H
{\displaystyle H}
with eigenvalue
λ
−
2
k
{\displaystyle \lambda -2k}
. Since, again,
H
{\displaystyle H}
has only finitely many eigenvectors, we conclude
that
v
l
{\displaystyle v_{l}}
must be zero for some
l
{\displaystyle l}
(and then
v
k
=
0
{\displaystyle v_{k}=0}
for all
k
>
l
{\displaystyle k>l}
).
Let
v
m
{\displaystyle v_{m}}
be the last nonzero vector 
in the chain; that is,
v
m
≠
0
{\displaystyle v_{m}\neq 0}
but
v
m
+
1
=
0
{\displaystyle v_{m+1}=0}
. Then of course
X
⋅
v
m
+
1
=
0
{\displaystyle X\cdot v_{m+1}=0}
and by the above identity with
k
=
m
+
1
{\displaystyle k=m+1}
, we have
0
=
X
⋅
v
m
+
1
=
(
m
+
1
)
(
λ
−
m
)
v
m
{\displaystyle 0=X\cdot v_{m+1}=(m+1)(\lambda
-m)v_{m}}
.Since
m
+
1
{\displaystyle m+1}
is at least one and
v
m
≠
0
{\displaystyle v_{m}\neq 0}
, we conclude that
λ
{\displaystyle \lambda }
must be equal to the non-negative integer
m
{\displaystyle m}
.
We thus obtain a chain of
m
+
1
{\displaystyle m+1}
vectors
v
0
,
…
,
v
m
{\displaystyle v_{0},\ldots ,v_{m}}
such that
Y
{\displaystyle Y}
acts as
Y
⋅
v
m
=
0
;
Y
⋅
v
k
=
v
k
+
1
(
k
<
m
)
{\displaystyle Y\cdot v_{m}=0;\quad Y\cdot
v_{k}=v_{k+1}\quad (k<m)}
and
X
{\displaystyle X}
acts as
X
⋅
v
0
=
0
,
X
⋅
v
k
=
k
(
m
−
(
k
−
1
)
)
v
k
−
1
(
k
>
0
)
{\displaystyle X\cdot v_{0}=0,\quad X\cdot
v_{k}=k(m-(k-1))v_{k-1}\quad (k>0)}
and
H
{\displaystyle H}
acts as
H
⋅
v
k
=
(
m
−
2
k
)
v
k
{\displaystyle H\cdot v_{k}=(m-2k)v_{k}}
.(We have replaced
λ
{\displaystyle \lambda }
with its currently known value of
m
{\displaystyle m}
in the above formulas.)
Since the vectors
v
k
{\displaystyle v_{k}}
are eigenvectors for
H
{\displaystyle H}
with distinct eigenvalues, they must be linearly
independent. Furthermore, the span of
v
0
,
…
,
v
m
{\displaystyle v_{0},\ldots ,v_{m}}
is clearly invariant under the action of the
complexified Lie algebra. Since
V
{\displaystyle V}
is assumed irreducible, this span must be
all of
V
{\displaystyle V}
. 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
m
≥
0
{\displaystyle m\geq 0}
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
m
{\displaystyle m}
, there is a unique irreducible representation
with highest weight
m
{\displaystyle m}
. Each irreducible representation is equivalent
to one of these. The representation with highest
weight
m
{\displaystyle m}
has dimension
m
+
1
{\displaystyle m+1}
with weights
m
,
m
−
2
,
…
,
−
(
m
−
2
)
,
−
m
{\displaystyle m,m-2,\ldots ,-(m-2),-m}
, each having multiplicity one.
=== The Casimir element ===
We now introduce the (quadratic) Casimir element,
C
{\displaystyle C}
given by
C
=
−
(
u
1
2
+
u
2
2
+
u
3
2
)
{\displaystyle C=-(u_{1}^{2}+u_{2}^{2}+u_{3}^{2})}
.We can view
C
{\displaystyle C}
as an element of the universal enveloping
algebra or as an operator in each irreducible
representation. Viewing
C
{\displaystyle C}
as an operator on the representation with
highest weight
m
{\displaystyle m}
, we may easily compute that
C
{\displaystyle C}
commutes with each
u
i
{\displaystyle u_{i}}
. Thus, by Schur's lemma,
C
{\displaystyle C}
acts as a scalar multiple
c
m
{\displaystyle c_{m}}
of the identity for each
m
{\displaystyle m}
. We can write
C
{\displaystyle C}
in terms of the
H
,
X
,
Y
{\displaystyle {H,X,Y}}
basis as follows:
C
=
(
X
+
Y
)
2
−
(
−
X
+
Y
)
2
+
H
2
{\displaystyle C=(X+Y)^{2}-(-X+Y)^{2}+H^{2}}
,which simplifies to
C
=
4
Y
X
+
H
2
+
2
H
{\displaystyle C=4YX+H^{2}+2H}
.The eigenvalue of
C
{\displaystyle C}
in the 
representation with highest weight
m
{\displaystyle m}
can be computed by applying
C
{\displaystyle C}
to the highest weight vector, which is annihilated
by
X
{\displaystyle X}
. Thus, we get
c
m
=
m
2
+
2
m
=
m
(
m
+
2
)
{\displaystyle c_{m}=m^{2}+2m=m(m+2)}
.In the physics literature, the Casimir is
normalized as
C
′
=
C
/
4
{\displaystyle C'=C/4}
. Labeling things in terms of
l
=
m
/
2
{\displaystyle l=m/2}
, the eigenvalue
d
l
{\displaystyle d_{l}}
of
C
′
{\displaystyle C'}
is then computed as
d
l
=
1
4
(
2
l
)
(
2
l
+
2
)
=
l
(
l
+
1
)
{\displaystyle d_{l}={\frac {1}{4}}(2l)(2l+2)=l(l+1)}
.
== 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
m
{\displaystyle m}
, we let
V
m
{\displaystyle V_{m}}
denote the space of homogeneous polynomials
of degree
m
{\displaystyle m}
in two complex variables. Then the dimension
of
V
m
{\displaystyle V_{m}}
is
m
+
1
{\displaystyle m+1}
. There is a natural action of SU(2) on each
V
m
{\displaystyle V_{m}}
, given by
[
U
⋅
p
]
(
z
)
=
p
(
U
−
1
z
)
,
z
∈
C
2
,
U
∈
S
U
(
2
)
{\displaystyle [U\cdot p](z)=p(U^{-1}z),\quad
z\in \mathbb {C} ^{2},\,U\in \mathrm {SU}
(2)}
.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
Π
:
G
→
G
L
(
V
)
{\displaystyle \Pi :G\rightarrow \mathrm {GL}
(V)}
is the function
X
:
G
→
C
{\displaystyle \mathrm {X} :G\rightarrow \mathbb
{C} }
given by
X
(
g
)
=
t
r
a
c
e
(
Π
(
g
)
)
{\displaystyle \mathrm {X} (g)=\mathrm {trace}
(\Pi (g))}
.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
T
{\displaystyle T}
consisting of the diagonal matrices in SU(2).
Since the irreducible representation with
highest weight
m
{\displaystyle m}
has weights
m
,
m
−
2
,
…
−
(
m
−
2
)
,
−
m
{\displaystyle m,m-2,\ldots -(m-2),-m}
, it is easy to see that the associated character
satisfies
X
(
(
e
i
θ
0
0
e
−
i
θ
)
)
=
e
i
m
θ
+
e
i
(
m
−
2
)
θ
+
⋯
e
−
i
(
m
−
2
)
θ
+
e
−
i
m
θ
.
{\displaystyle \mathrm {X} \left({\begin{pmatrix}e^{i\theta
}&0\\0&e^{-i\theta }\end{pmatrix}}\right)=e^{im\theta
}+e^{i(m-2)\theta }+\cdots e^{-i(m-2)\theta
}+e^{-im\theta }.}
This expression is a finite geometric series
that can be simplified to
X
(
(
e
i
θ
0
0
e
−
i
θ
)
)
=
s
i
n
(
(
m
+
1
)
θ
)
s
i
n
(
θ
)
.
{\displaystyle \mathrm {X} \left({\begin{pmatrix}e^{i\theta
}&0\\0&e^{-i\theta }\end{pmatrix}}\right)={\frac
{\mathrm {sin} ((m+1)\theta )}{\mathrm {sin}
(\theta )}}.}
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
m
{\displaystyle m}
is even) or all of the weights are odd (if
m
{\displaystyle m}
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,
m
{\displaystyle m}
being even corresponds to
l
{\displaystyle l}
being an integer while
m
{\displaystyle m}
being odd corresponds to
l
{\displaystyle l}
being a half-integer. These two cases are
described as integer spin and half-integer
spin, respectively. The representations with
odd, positive values of
m
{\displaystyle m}
are faithful representations of SU(2), while
the representations of SU(2) with non-negative,
even
m
{\displaystyle m}
are not faithful.
== Another approach ==
See under the example for Borel–Weil–Bott
theorem.
== Most important irreducible representations
and their applications ==
As stated above, 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
m
=
1
{\displaystyle m=1}
(i.e.,
l
=
1
/
2
{\displaystyle l=1/2}
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
2 × 2 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
m
=
2
{\displaystyle m=2}
(i.e.,
l
=
1
{\displaystyle l=1}
) 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
m
=
3
{\displaystyle m=3}
(i.e.
l
=
3
/
2
{\displaystyle l=3/2}
) representation is used in particle physics
for certain baryons, such as the Δ.
== See also ==
Rotation operator (vector space)
Rotation operator (quantum mechanics)
Representation theory of SO(3)
Connection between SO(3) and SU(2)
representation theory of SL2(R)
Electroweak interaction
Rotation group SO(3) § A note on Lie algebra
