Editing Representation theory of SU(2) (section)

==Lie algebra representations==
{{see also|Special linear Lie algebra#Representation theory}}
The representations of the group are found by considering representations of <math>\mathfrak{su}(2)</math>, the [[Special unitary group#Lie algebra basis|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;<ref>{{harvnb|Hall|2015}} Theorem 5.6</ref> we will give an explicit construction of the representations at the group level below.<ref>{{Harv|Hall|2015}}, Section 4.6</ref>

===Real and complexified Lie algebras===
The real Lie algebra <math>\mathfrak{su}(2)</math> has a [[Special unitary group#Lie algebra basis|basis given by]]
:<math>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}~,
</math>
(These basis matrices are related to the [[Pauli matrices]] by <math>u_1 = +i\ \sigma_1 \;, \, u_2 = -i\ \sigma_2 \;,</math>  and <math>u_3 = +i\ \sigma_3 ~.</math>)

The matrices are a representation of the [[quaternion]]s:
:<math> u_1\,u_1 = -I\, , ~~\quad u_2\,u_2 = -I \, , ~~\quad u_3\,u_3 = -I\, ,</math>
:<math> u_1\,u_2 = +u_3\, , \quad u_2\,u_3 = +u_1\, , \quad u_3\,u_1 = +u_2\, ,</math>
:<math> u_2\,u_1 = -u_3\, , \quad u_3\,u_2 = -u_1\, , \quad u_1\,u_3 = -u_2 ~.</math>

where {{mvar|I}} is the conventional 2×2 identity matrix:<math>~~I = \begin{bmatrix}
  1 & 0\\
  0 & 1
\end{bmatrix} ~.</math>

Consequently, the [[commutator|commutator brackets]] of the matrices satisfy
:<math>[u_1, u_2] = 2 u_3\, ,\quad [u_2, u_3] = 2 u_1\, ,\quad [u_3, u_1] = 2 u_2 ~.</math>

It is then convenient to pass to the complexified Lie algebra
:<math>\mathfrak{su}(2) + i\,\mathfrak{su}(2) = \mathfrak{sl}(2;\mathbb C) ~.</math>

(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 <math>\mathbb C</math> this passage from real to complexified Lie algebra is harmless.<ref>{{harvnb|Hall|2015}}, Section&nbsp;3.6</ref> 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 <math>\mathfrak{su}(2)</math>.

The complexified Lie algebra is spanned by three elements <math>X</math>, <math>Y</math>, and <math>H</math>, given by
:<math>
 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) ~;
</math>

or, explicitly, 
:<math> 
  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}
~.</math>

The non-trivial/non-identical part of the group's multiplication table is
:<math> H X ~=~~~~X ,\qquad H Y ~=   -Y ,\qquad X Y ~=~ \tfrac{1}{2}\left(I + H \right),</math>
:<math> X H ~=   -X ,\qquad Y H ~=~~~~Y ,\qquad Y X ~=~ \tfrac{1}{2}\left(I - H \right),</math>
:<math> H H ~=~~~I~ ,\qquad X X ~=~~~~O ,\qquad Y Y ~=~ ~O,</math>

where {{mvar|O}} is the 2×2 all-zero matrix.
Hence their commutation relations are
:<math>[H, X] = 2 X, \qquad [H, Y] = -2 Y, \qquad [X, Y] = H.</math>

Up to a factor of 2, the elements <math>H</math>, <math>X</math> and <math>Y</math> may be identified with the angular momentum operators <math>J_z</math>, <math>J_+</math>, and <math>J_-</math>, 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 <math>H</math> are referred to as the '''weights''' of the representation. The following elementary result<ref>{{harvnb|Hall|2015}} Lemma 4.33</ref> is a key step in the analysis. Suppose that <math>v</math> is an [[eigenvector]] for <math>H</math> with eigenvalue <math>\alpha</math>; that is, that <math>H v = \alpha v.</math> Then
:<math>\begin{alignat}{5}
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{alignat}</math>

In other words, <math>Xv</math> is either the zero vector or an eigenvector for <math>H</math> with eigenvalue <math>\alpha + 2</math> and <math>Y v</math> is either zero or an eigenvector for <math>H</math> with eigenvalue <math>\alpha - 2.</math> Thus, the operator <math>X</math> acts as a '''raising operator''', increasing the weight by 2, while <math>Y</math> acts as a '''lowering operator'''.

Suppose now that <math>V</math> is an irreducible, finite-dimensional representation of the [[complexification|complexified]] Lie algebra. Then <math>H</math> can have only finitely many eigenvalues. In particular, there must be some final eigenvalue <math>\lambda \in \mathbb{C}</math> with the property that <math>\lambda + 2</math> is ''not'' an eigenvalue. Let <math>v_0</math> be an eigenvector for <math>H</math> with that eigenvalue <math>\lambda:</math> 

:<math>H v_0 = \lambda v_0,</math>

then we must have

:<math>X v_0 = 0,</math>

or else the above identity would tell us that <math>X v_0</math> is an eigenvector with eigenvalue <math>\lambda + 2  .</math>

Now define a "chain" of vectors <math>v_0, v_1, \ldots</math> by
:<math>v_k = Y^k  v_0</math>.

A simple argument by [[mathematical induction|induction]]<ref>{{harvnb|Hall|2015}}, Equation (4.15)</ref> then shows that
:<math>X  v_k = k(\lambda - (k - 1))v_{k-1}</math> 
for all <math>k = 1, 2, \ldots .</math> Now, if <math> v_k </math> is not the zero vector, it is an eigenvector for <math>H</math> with eigenvalue <math> \lambda - 2k .</math> Since, again, <math> H </math> has only finitely many eigenvectors, we conclude that <math> v_\ell </math> must be zero for some <math> \ell </math> (and then <math>v_k = 0</math> for all <math> k > \ell </math>).

Let <math>v_m</math> be the last nonzero vector in the chain; that is, <math> v_m \neq 0 </math> but <math> v_{m+1} = 0 .</math> Then of course <math> X  v_{m+1} = 0 </math> and by the above identity with <math>k = m + 1 ,</math> we have
:<math> 0 = X  v_{m+1} = (m + 1)(\lambda - m)v_m .</math>

Since <math> m + 1 </math> is at least one and <math> v_m \neq 0  ,</math> we conclude that <math> \lambda </math> ''must be equal to the non-negative integer'' <math> m  .</math>

We thus obtain a chain of <math> m + 1 </math> vectors, <math> v_0, v_1, \ldots, v_m  ,</math> such that <math> Y </math> acts as
:<math> Y  v_m = 0, \quad Y  v_k = v_{k+1} \quad (k < m) </math>
and <math> X </math> acts as
:<math> X  v_0 = 0, \quad X  v_k = k  (m - (k - 1))  v_{k-1} \quad (k \ge 1)</math>
and <math> H </math> acts as
:<math>H  v_k = (m - 2k)  v_k .</math>
(We have replaced <math>\lambda</math> with its currently known value of <math> m </math> in the formulas above.)

Since the vectors <math> v_k </math> are eigenvectors for <math>H</math> with distinct eigenvalues, they must be linearly independent. Furthermore, the span of <math> v_0, \ldots , v_m </math> is clearly invariant under the action of the complexified Lie algebra. Since <math>V</math> is assumed irreducible, this span must be all of <math> V .</math> 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 <math> m \geq 0 </math> 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.<ref>{{harvnb|Hall|2015}}, proof of Proposition&nbsp;4.11</ref>

'''Conclusion''': For each non-negative integer <math> m ,</math> there is a unique irreducible representation with highest weight <math> m  .</math> Each irreducible representation is equivalent to one of these. The representation with highest weight <math> m </math> has dimension <math> m + 1 </math> with weights <math> m, m - 2, \ldots, -(m - 2), -m  ,</math> each having multiplicity one.

===The Casimir element===
We now introduce the (quadratic) [[Casimir element]], <math>C</math> given by
:<math>C = -\left(u_1^2 + u_2^2 + u_3^2\right)</math>.

We can view <math>C</math> as an element of the [[universal enveloping algebra]] or as an operator in each irreducible representation. Viewing <math> C </math> as an operator on the representation with highest weight <math> m </math>, we may easily compute that <math> C </math> commutes with each <math> u_i  .</math> Thus, by [[Schur's lemma]], <math> C </math> acts as a scalar multiple <math>c_m</math> of the identity for each <math> m .</math> We can write <math>C</math> in terms of the <math>\{ H, X, Y \} </math> basis as follows:
:<math>C = (X + Y)^2 - (-X + Y)^2 + H^2  ,</math>
which can be reduced to
:<math>C = 4YX + H^2 + 2H .</math>

The eigenvalue of <math> C </math> in the representation with highest weight <math> m </math> can be computed by applying <math> C </math> to the highest weight vector, which is annihilated by <math> X  ;</math> thus, we get 
:<math>c_m = m^2 + 2m = m(m + 2) .</math>

In the physics literature, the Casimir is normalized as <math display="inline"> C' = \frac{1}{4}C .</math> Labeling things in terms of <math display="inline"> \ell = \frac{1}{2}m  ,</math> the eigenvalue <math> d_\ell </math> of <math> C' </math> is then computed as
:<math> d_\ell = \frac{1}{4}(2\ell)(2\ell + 2) = \ell (\ell + 1) .</math>