Editing Representation theory of SU(2) (section)

===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.