Real and complexified Lie algebras

The real Lie algebra ${\displaystyle {\mathfrak {su}}(2)}$ has a basis given by

${\displaystyle u_{1}={\begin{bmatrix}0&i\\i&0\end{bmatrix}},\qquad u_{2}={\begin{bmatrix}0&-1\\1&~~0\end{bmatrix}},\qquad u_{3}={\begin{bmatrix}i&~~0\\0&-i\end{bmatrix}}~,}$

(These basis matrices are related to the Pauli matrices by ${\displaystyle u_{1}=+i\ \sigma _{1}\;,\,u_{2}=-i\ \sigma _{2}\;,}$ and ${\displaystyle u_{3}=+i\ \sigma _{3}~.}$)

The matrices are a representation of the quaternions:

${\displaystyle u_{1}\,u_{1}=-I\,,~~\quad u_{2}\,u_{2}=-I\,,~~\quad u_{3}\,u_{3}=-I\,,}$

${\displaystyle u_{1}\,u_{2}=+u_{3}\,,\quad u_{2}\,u_{3}=+u_{1}\,,\quad u_{3}\,u_{1}=+u_{2}\,,}$

${\displaystyle u_{2}\,u_{1}=-u_{3}\,,\quad u_{3}\,u_{2}=-u_{1}\,,\quad u_{1}\,u_{3}=-u_{2}~.}$

where I is the conventional 2×2 identity matrix:${\displaystyle ~~I={\begin{bmatrix}1&0\\0&1\end{bmatrix}}~.}$

Consequently, the commutator brackets of the matrices satisfy

${\displaystyle [u_{1},u_{2}]=2u_{3}\,,\quad [u_{2},u_{3}]=2u_{1}\,,\quad [u_{3},u_{1}]=2u_{2}~.}$

It is then convenient to pass to the complexified Lie algebra

${\displaystyle \mathrm {su} (2)+i\,\mathrm {su} (2)=\mathrm {sl} (2;\mathbb {C} )~.}$

(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 ${\displaystyle \mathbb {C} }$ this passage from real to complexified Lie algebra is harmless.^[3] 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 ${\displaystyle {\mathfrak {su}}(2)}$.

The complexified Lie algebra is spanned by three elements ${\displaystyle X}$, ${\displaystyle Y}$, and ${\displaystyle H}$, given by

${\displaystyle H={\frac {1}{i}}u_{3},\qquad X={\frac {1}{2i}}\left(u_{1}-iu_{2}\right),\qquad Y={\frac {1}{2i}}(u_{1}+iu_{2})~;}$

or, explicitly,

${\displaystyle H={\begin{bmatrix}1&~~0\\0&-1\end{bmatrix}},\qquad X={\begin{bmatrix}0&1\\0&0\end{bmatrix}},\qquad Y={\begin{bmatrix}0&0\\1&0\end{bmatrix}}~.}$

The non-trivial/non-identical part of the group's multiplication table is

${\displaystyle H\,X~=~~~~X,\qquad H\,Y~=-Y,\qquad X\,Y~=~{\tfrac {1}{2}}\left(I+H\right),}$

${\displaystyle X\,H~=-X,\qquad Y\,H~=~~~~Y,\qquad Y\,X~=~{\tfrac {1}{2}}\left(I-H\right),}$

${\displaystyle H\,H~=~~~I~,\qquad X\,X~=~~~~O,\qquad Y\,Y~=~~O~~;}$

where O is the 2×2 all-zero matrix. Hence their commutation relations are

${\displaystyle [H,X]=2\,X\,,\qquad [H,Y]=-2\,Y\,,\qquad [X,Y]=H~.}$

Up to a factor of 2, the elements ${\displaystyle H}$, ${\displaystyle X}$ and ${\displaystyle Y}$ may be identified with the angular momentum operators ${\displaystyle J_{z}}$, ${\displaystyle J_{+}}$, and ${\displaystyle J_{-}}$, 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 ${\displaystyle H}$ are referred to as the weights of the representation. The following elementary result^[4] is a key step in the analysis. Suppose that ${\displaystyle v}$ is an eigenvector for ${\displaystyle H}$ with eigenvalue ${\displaystyle \;\alpha \;;}$ that is, that ${\displaystyle \;H\,v=\alpha \,v~.}$ Then

${\displaystyle {\begin{aligned}H\,(X\,v)~=~(\,X\,H+[H,X]\,)\,v~&=~(\alpha +2)\,X\,v\;,\\[3pt]H\,(Y\,v)~=~(\,Y\,H+\,[H,Y]\,)\,v~&=~(\alpha -2)\,Y\,v~.\end{aligned}}}$

In other words, ${\displaystyle \,X\,v\,}$ is either the zero vector or an eigenvector for ${\displaystyle \,H\,}$ with eigenvalue ${\displaystyle \,\alpha +2\,}$ and ${\displaystyle \,Y\,v\,}$ is either zero or an eigenvector for ${\displaystyle \,H\,}$ with eigenvalue ${\displaystyle \,\alpha -2~.}$ Thus, the operator ${\displaystyle \,X\,}$ acts as a raising operator, increasing the weight by 2, while ${\displaystyle \,Y\,}$ acts as a lowering operator.

Suppose now that ${\displaystyle \,V\,}$ is an irreducible, finite-dimensional representation of the complexified Lie algebra. Then ${\displaystyle \,H\,}$ can have only finitely many eigenvalues. In particular, there must be some final eigenvalue ${\displaystyle \;\lambda \in \mathbb {C} \;}$ with the property that ${\displaystyle \,\lambda +2\,}$ is not an eigenvalue. Let ${\displaystyle \,v_{0}\,}$ be an eigenvector for ${\displaystyle \,H\,}$ with that eigenvalue ${\displaystyle \,\lambda \;:}$

${\displaystyle H\,v_{0}=\lambda \,v_{0}\;,}$

then we must have

${\displaystyle X\,v_{0}=0\;,}$

or else the above identity would tell us that ${\displaystyle \,X\,v_{0}\,}$ is an eigenvector with eigenvalue ${\displaystyle \,\lambda +2\;.}$

Now define a "chain" of vectors ${\displaystyle v_{0},v_{1},\ldots }$ by

${\displaystyle v_{k}=Y^{k}\,v_{0}}$.

A simple argument by induction^[5] then shows that

${\displaystyle X\,v_{k}=k\,(\lambda -(k-1))\,v_{k-1}}$

for all ${\displaystyle k=1,2,\ldots ~.}$ Now, if ${\displaystyle \,v_{k}\,}$ is not the zero vector, it is an eigenvector for ${\displaystyle H}$ with eigenvalue ${\displaystyle \,\lambda -2k~.}$ Since, again, ${\displaystyle \,H\,}$ has only finitely many eigenvectors, we conclude that ${\displaystyle \,v_{\ell }\,}$ must be zero for some ${\displaystyle \,\ell \,}$ (and then ${\displaystyle v_{k}=0}$ for all ${\displaystyle \,k>\ell \,}$).

Let ${\displaystyle v_{m}}$ be the last nonzero vector in the chain; that is, ${\displaystyle \;v_{m}\neq 0\;}$ but ${\displaystyle \;v_{m+1}=0~.}$ Then of course ${\displaystyle \;X\,v_{m+1}=0\;}$ and by the above identity with ${\displaystyle k=m+1\;,}$ we have

${\displaystyle \;0=X\,v_{m+1}=(m+1)\,(\lambda -m)\,v_{m}~.}$

Since ${\displaystyle \,m+1\,}$ is at least one and ${\displaystyle \,v_{m}\neq 0\;,}$ we conclude that ${\displaystyle \,\lambda \,}$ must be equal to the non-negative integer ${\displaystyle \,m\;.}$

We thus obtain a chain of ${\displaystyle \,m+1\,}$ vectors, ${\displaystyle \;v_{0},v_{1},\ldots ,v_{m}\;,}$ such that ${\displaystyle \,Y\,}$ acts as

${\displaystyle Y\,v_{m}=0,\quad Y\,v_{k}=v_{k+1}\quad (k<m)}$

and ${\displaystyle \,X\,}$ acts as

${\displaystyle X\,v_{0}=0,\quad X\,v_{k}=k\,(m-(k-1))\,v_{k-1}\quad (k\geq 1)}$

and ${\displaystyle \,H\,}$ acts as

${\displaystyle H\,v_{k}=(m-2k)\,v_{k}~.}$

(We have replaced ${\displaystyle \lambda }$ with its currently known value of ${\displaystyle \,m\,}$ in the formulas above.)

Since the vectors ${\displaystyle \,v_{k}\,}$ are eigenvectors for ${\displaystyle H}$ with distinct eigenvalues, they must be linearly independent. Furthermore, the span of ${\displaystyle \;v_{0},\ldots ,v_{m}\;}$ is clearly invariant under the action of the complexified Lie algebra. Since ${\displaystyle V}$ is assumed irreducible, this span must be all of ${\displaystyle \,V\;.}$ 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 ${\displaystyle \;m\geq 0\;}$ 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.^[6]

Conclusion: For each non-negative integer ${\displaystyle \,m\,,}$ there is a unique irreducible representation with highest weight ${\displaystyle \,m\;.}$ Each irreducible representation is equivalent to one of these. The representation with highest weight ${\displaystyle \,m\,}$ has dimension ${\displaystyle \,m+1\,}$ with weights ${\displaystyle \;m,m-2,\ldots ,-(m-2),-m\;,}$ each having multiplicity one.

The Casimir element

We now introduce the (quadratic) Casimir element, ${\displaystyle C}$ given by

${\displaystyle C=-\left(u_{1}^{2}+u_{2}^{2}+u_{3}^{2}\right)}$.

We can view ${\displaystyle C}$ as an element of the universal enveloping algebra or as an operator in each irreducible representation. Viewing ${\displaystyle \,C\,}$ as an operator on the representation with highest weight ${\displaystyle \,m\,}$, we may easily compute that ${\displaystyle \,C\,}$ commutes with each ${\displaystyle \,u_{i}\;.}$ Thus, by Schur's lemma, ${\displaystyle \,C\,}$ acts as a scalar multiple ${\displaystyle c_{m}}$ of the identity for each ${\displaystyle \,m\;.}$ We can write ${\displaystyle C}$ in terms of the ${\displaystyle \,\{H,X,Y\}\,}$ basis as follows:

${\displaystyle C=(X+Y)^{2}-(-X+Y)^{2}+H^{2}\;,}$

which can be reduced to

${\displaystyle C=4YX+H^{2}+2H~.}$

The eigenvalue of ${\displaystyle \,C\,}$ in the representation with highest weight ${\displaystyle \,m\,}$ can be computed by applying ${\displaystyle \,C\,}$ to the highest weight vector, which is annihilated by ${\displaystyle \,X\;;}$ thus, we get

${\displaystyle c_{m}=m^{2}+2m=m(m+2)~.}$

In the physics literature, the Casimir is normalized as ${\textstyle \;C'={\frac {1}{4}}\,C~.}$ Labeling things in terms of ${\textstyle \;\ell ={\frac {1}{2}}\,m\;,}$ the eigenvalue ${\displaystyle \,d_{\ell }\,}$ of ${\displaystyle \,C'\,}$ is then computed as

${\displaystyle d_{\ell }={\frac {1}{4}}\,(2\ell )\,(2\ell +2)=\ell \,(\ell +1)~.}$

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.^[7] That is, for each non-negative integer ${\displaystyle m}$, we let ${\displaystyle V_{m}}$ denote the space of homogeneous polynomials ${\displaystyle p}$ of degree ${\displaystyle m}$ in two complex variables. Then the dimension of ${\displaystyle V_{m}}$ is ${\displaystyle m+1}$. There is a natural action of SU(2) on each ${\displaystyle V_{m}}$, given by

${\displaystyle [U\cdot p](z)=p\left(U^{-1}z\right),\quad z\in \mathbb {C} ^{2},\,U\in \mathrm {SU} (2)}$.

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 ${\displaystyle \Pi :G\rightarrow \operatorname {GL} (V)}$ is the function ${\displaystyle \mathrm {X} :G\rightarrow \mathbb {C} }$ given by

${\displaystyle \mathrm {X} (g)=\operatorname {trace} (\Pi (g))}$.

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 ${\displaystyle T}$ consisting of the diagonal matrices in SU(2), since the elements are orthogonally diagonalizable with the spectral theorem.^[8] Since the irreducible representation with highest weight ${\displaystyle m}$ has weights ${\displaystyle m,m-2,\ldots ,-(m-2),-m}$, it is easy to see that the associated character satisfies

${\displaystyle \mathrm {X} \left({\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}\right)=e^{im\theta }+e^{i(m-2)\theta }+\cdots +e^{-i(m-2)\theta }+e^{-im\theta }.}$

This expression is a finite geometric series that can be simplified to

${\displaystyle \mathrm {X} \left({\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}\right)={\frac {\sin((m+1)\theta )}{\sin(\theta )}}.}$

This last expression is just the statement of the Weyl character formula for the SU(2) case.^[9]

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 ${\displaystyle m}$ is even) or all of the weights are odd (if ${\displaystyle m}$ is odd). In physical terms, this distinction is important: The representations with even weights correspond to ordinary representations of the rotation group SO(3).^[10] 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, ${\displaystyle m}$ being even corresponds to ${\displaystyle l}$ being an integer while ${\displaystyle m}$ being odd corresponds to ${\displaystyle l}$ being a half-integer. These two cases are described as integer spin and half-integer spin, respectively. The representations with odd, positive values of ${\displaystyle m}$ are faithful representations of SU(2), while the representations of SU(2) with non-negative, even ${\displaystyle m}$ are not faithful.^[11]