Editing 3D rotation group (section)

===A note on Lie algebras===
{{Main|Angular momentum operator}}
{{see also|Representation theory of SU(2)|Jordan map}}

In [[Lie algebra representation]]s, the group SO(3) is compact and simple of rank 1, and so it has a single independent [[Casimir element]], a quadratic invariant function of the three generators which commutes with all of them. The Killing form for the rotation group is just the [[Kronecker delta]], and so this Casimir invariant is simply the sum of the squares of the generators, <math>\boldsymbol{J}_x, \boldsymbol{J}_y, \boldsymbol{J}_z,</math> of the algebra
:<math>
[\boldsymbol{J}_x, \boldsymbol{J}_y] = \boldsymbol{J}_z, \quad
[\boldsymbol{J}_z, \boldsymbol{J}_x] = \boldsymbol{J}_y, \quad
[\boldsymbol{J}_y, \boldsymbol{J}_z] = \boldsymbol{J}_x.
</math>
That is, the Casimir invariant is given by
:<math>\boldsymbol{J}^2\equiv \boldsymbol{J}\cdot \boldsymbol{J} =\boldsymbol{J}_x^2+\boldsymbol{J}_y^2+\boldsymbol{J}_z^2 \propto \boldsymbol{I}.</math>

For unitary irreducible [[Lie algebra representation|representations]] {{mvar|D<sup>j</sup>}}, the eigenvalues of this invariant are real and discrete, and characterize each representation, which is finite dimensional, of dimensionality <math>2j+1</math>. That is, the eigenvalues of this Casimir operator are
:<math>\boldsymbol{J}^2=- j(j+1) \boldsymbol{I}_{2j+1},</math>
where {{mvar|j}} is integer or half-integer, and referred to as the [[Spin (physics)|spin]] or [[angular momentum]].

So, the 3 × 3 generators '''''L''''' displayed above act on the triplet (spin 1) representation, while the 2 × 2 generators below, '''''t''''', act on the [[Spinor|doublet]] ([[spin-1/2]]) representation. By taking [[Kronecker product]]s of {{math|''D''<sup>1/2</sup>}} with itself repeatedly, one may construct all higher irreducible representations {{mvar|D<sup>j</sup>}}. That is, the resulting generators for higher spin systems in three spatial dimensions, for arbitrarily large {{mvar|j}}, can be calculated using these [[spin operator]]s and [[ladder operator]]s.

For every unitary irreducible representations {{mvar|D<sup>j</sup>}} there is an equivalent one, {{math|''D''<sup>−''j''−1</sup>}}. All 
infinite-dimensional irreducible representations must be non-unitary, since the group is compact.

In [[quantum mechanics]], the Casimir invariant is the "angular-momentum-squared" operator; integer values of spin {{mvar|j}} characterize [[boson|bosonic representation]]s, while half-integer values [[fermion|fermionic representation]]s. The [[Skew-Hermitian matrix|antihermitian]] matrices used above are utilized as [[spin operator]]s, after they are multiplied by {{mvar|i}}, so they are now [[Hermitian matrix|hermitian]] (like the Pauli matrices). Thus, in this language,
:<math>
[\boldsymbol{J}_x, \boldsymbol{J}_y] = i\boldsymbol{J}_z, \quad
[\boldsymbol{J}_z, \boldsymbol{J}_x] = i\boldsymbol{J}_y, \quad
[\boldsymbol{J}_y, \boldsymbol{J}_z] = i\boldsymbol{J}_x.
</math>
and hence
:<math>\boldsymbol{J}^2= j(j+1) \boldsymbol{I}_{2j+1}.</math>

Explicit expressions for these {{mvar|D<sup>j</sup>}} are,
:<math>\begin{align} 
\left (\boldsymbol{J}_z^{(j)}\right )_{ba} &= (j+1-a)\delta_{b,a}\\
\left (\boldsymbol{J}_x^{(j)}\right )_{ba} &=\frac{1}{2}  \left (\delta_{b,a+1}+\delta_{b+1,a} \right ) \sqrt{(j+1)(a+b-1)-ab}\\
\left (\boldsymbol{J}_y^{(j)}\right )_{ba} &=\frac{1}{2i} \left (\delta_{b,a+1}-\delta_{b+1,a} \right ) \sqrt{(j+1)(a+b-1)-ab}\\
\end{align}</math>
where {{mvar|j}} is arbitrary and <math>1 \le a, b \le 2j+1</math>.

For example, the resulting spin matrices for spin 1 (<math>j = 1</math>) are
:<math>\begin{align}
\boldsymbol{J}_x &= \frac{1}{\sqrt{2}}
 \begin{pmatrix}
 0 &1 &0\\
 1 &0 &1\\
 0 &1 &0
 \end{pmatrix} \\
\boldsymbol{J}_y &= \frac{1}{\sqrt{2}}
 \begin{pmatrix}
 0 &-i &0\\
 i &0 &-i\\
 0 &i &0
 \end{pmatrix} \\
\boldsymbol{J}_z &=
 \begin{pmatrix}
 1 &0 &0\\
 0 &0 &0\\
 0 &0 &-1
 \end{pmatrix}
\end{align}</math>

Note, however, how these are in an equivalent, but different basis, the [[Spherical basis#Change of basis matrix|spherical basis]], than the above  {{mvar|i}}'''''L'''''  in the Cartesian basis.<ref group="nb">Specifically, <math>\boldsymbol{U} \boldsymbol{J}_{\alpha}\boldsymbol{U}^\dagger=i\boldsymbol{L}_\alpha</math> for 

: <math>\boldsymbol{U}= \left(
\begin{array}{ccc}
 -\frac{i}{\sqrt{2}} & 0 & \frac{i}{\sqrt{2}} \\
 \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\
 0 & i & 0 \\
\end{array}
\right).</math></ref>

For  higher spins,  such as spin {{sfrac|3|2}} (<math>j=\tfrac{3}{2}</math>):
:<math>\begin{align}
\boldsymbol{J}_x &= \frac{1}{2}
 \begin{pmatrix}
 0 &\sqrt{3} &0 &0\\
 \sqrt{3} &0 &2 &0\\
 0 &2 &0 &\sqrt{3}\\
 0 &0 &\sqrt{3} &0
 \end{pmatrix} \\
\boldsymbol{J}_y &= \frac{1}{2}
 \begin{pmatrix}
 0 &-i\sqrt{3} &0 &0\\
 i\sqrt{3} &0 &-2i &0\\
 0 &2i &0 &-i\sqrt{3}\\
 0 &0 &i\sqrt{3} &0
 \end{pmatrix} \\
\boldsymbol{J}_z &=\frac{1}{2}
 \begin{pmatrix}
 3 &0 &0 &0\\
 0 &1 &0 &0\\
 0 &0 &-1 &0\\
 0 &0 &0 &-3
 \end{pmatrix}.
\end{align}</math>

For spin {{sfrac|5|2}} (<math>j = \tfrac{5}{2}</math>),
:<math>\begin{align}
\boldsymbol{J}_x &= \frac{1}{2}
 \begin{pmatrix}
 0 &\sqrt{5} &0 &0 &0 &0 \\
 \sqrt{5} &0 &2\sqrt{2} &0 &0 &0 \\
 0 &2\sqrt{2} &0 &3 &0 &0 \\
 0 &0 &3 &0 &2\sqrt{2} &0 \\
 0 &0 &0 &2\sqrt{2} &0 &\sqrt{5} \\
 0 &0 &0 &0 &\sqrt{5} &0
 \end{pmatrix} \\
\boldsymbol{J}_y &= \frac{1}{2}
 \begin{pmatrix}
 0 &-i\sqrt{5} &0 &0 &0 &0 \\
 i\sqrt{5} &0 &-2i\sqrt{2} &0 &0 &0 \\
 0 &2i\sqrt{2} &0 &-3i &0 &0 \\
 0 &0 &3i &0 &-2i\sqrt{2} &0 \\
 0 &0 &0 &2i\sqrt{2} &0 &-i\sqrt{5} \\
 0 &0 &0 &0 &i\sqrt{5} &0
 \end{pmatrix} \\
\boldsymbol{J}_z &= \frac{1}{2}
 \begin{pmatrix}
 5 &0 &0 &0 &0 &0 \\
 0 &3 &0 &0 &0 &0 \\
 0 &0 &1 &0 &0 &0 \\
 0 &0 &0 &-1 &0 &0 \\
 0 &0 &0 &0 &-3 &0 \\
 0 &0 &0 &0 &0 &-5
 \end{pmatrix}.
\end{align}</math>

{{main|Spin (physics)#Higher spins}}