Editing 3D rotation group (section)

==Lie algebra==
Associated with every [[Lie group]] is its [[Lie algebra]], a linear space of the same dimension as the Lie group, closed under a bilinear alternating product called the [[Lie bracket]]. The Lie algebra of {{math|SO(3)}} is denoted by <math>\mathfrak{so}(3)</math> and consists of all [[skew-symmetric matrix|skew-symmetric]] {{math|3 × 3}} matrices.<ref>{{harvnb|Hall|2015}} Proposition 3.24</ref> This may be seen by differentiating the [[orthogonal matrix|orthogonality condition]], {{math|1=''A''<sup>T</sup>''A'' = ''I'', ''A'' ∈ SO(3)}}.<ref group="nb">For an alternative derivation of <math>\mathfrak{so}(3)</math>, see [[Classical group]].</ref> The Lie bracket of two elements of <math>\mathfrak{so}(3)</math> is, as for the Lie algebra of every matrix group, given by the matrix [[commutator]], {{math|1=[''A''<sub>1</sub>, ''A''<sub>2</sub>] = ''A''<sub>1</sub>''A''<sub>2</sub> − ''A''<sub>2</sub>''A''<sub>1</sub>}}, which is again a skew-symmetric matrix. The Lie algebra bracket captures the essence of the Lie group product in a sense made precise by the [[Baker–Campbell–Hausdorff formula]].

The elements of <math>\mathfrak{so}(3)</math> are the "infinitesimal generators" of rotations, i.e., they are the elements of the [[tangent space]] of the manifold SO(3) at the identity element. If <math>R(\phi, \boldsymbol{n})</math> denotes a counterclockwise rotation with angle φ about the axis specified by the unit vector <math>\boldsymbol{n},</math> then

:<math>\forall \boldsymbol{u} \in \R^3: \qquad \left. \frac{\operatorname{d}}{\operatorname{d}\phi} \right|_{\phi=0} R(\phi,\boldsymbol{n}) \boldsymbol{u} = \boldsymbol{n} \times \boldsymbol{u}.</math>

This can be used to show that the Lie algebra <math>\mathfrak{so}(3)</math> (with commutator) is isomorphic to the Lie algebra <math>\R^3</math> (with [[cross product]]). Under this isomorphism, an [[Axis–angle representation#Rotation vector|Euler vector]] <math>\boldsymbol{\omega}\in\R^3</math> corresponds to the linear map <math>\widetilde{\boldsymbol{\omega}}</math> defined by <math>\widetilde{\boldsymbol{\omega}}(\boldsymbol{u}) = \boldsymbol{\omega}\times\boldsymbol{u}.</math>

In more detail, most often a suitable basis for <math>\mathfrak{so}(3)</math> as a {{nowrap|{{math|3}}-dimensional}} vector space is

:<math>
\boldsymbol{L}_x = \begin{bmatrix}0&0&0\\0&0&-1\\0&1&0\end{bmatrix}, \quad
\boldsymbol{L}_y = \begin{bmatrix}0&0&1\\0&0&0\\-1&0&0\end{bmatrix}, \quad
\boldsymbol{L}_z = \begin{bmatrix}0&-1&0\\1&0&0\\0&0&0\end{bmatrix}.
</math>

The [[commutation relation]]s of these basis elements are,

:<math>
[\boldsymbol{L}_x, \boldsymbol{L}_y] = \boldsymbol{L}_z, \quad
[\boldsymbol{L}_z, \boldsymbol{L}_x] = \boldsymbol{L}_y, \quad
[\boldsymbol{L}_y, \boldsymbol{L}_z] = \boldsymbol{L}_x
</math>

which agree with the relations of the three [[standard basis|standard unit vectors]] of <math>\R^3</math> under the cross product.

As announced above, one can identify any matrix in this Lie algebra with an Euler vector <math>\boldsymbol{\omega} = (x,y,z) \in \R^3,</math><ref>{{harvnb|Rossmann|2002}}</ref>

:<math>\widehat{\boldsymbol{\omega}} =\boldsymbol{\omega}\cdot \boldsymbol{L} = x \boldsymbol{L}_x + y \boldsymbol{L}_y + z \boldsymbol{L}_z =\begin{bmatrix}0&-z&y\\z&0&-x\\-y&x&0\end{bmatrix} \in \mathfrak{so}(3).</math>

This identification is sometimes called the '''hat-map'''.<ref name="Engø 2001">{{harvnb|Engø|2001}}</ref> Under this identification, the <math>\mathfrak{so}(3)</math> bracket corresponds in <math>\R^3</math> to the [[cross product]],

:<math>\left [\widehat{\boldsymbol{u}},\widehat{\boldsymbol{v}} \right ] = \widehat{\boldsymbol{u} \times \boldsymbol{v}}.</math>

The matrix identified with a vector <math>\boldsymbol{u}</math> has the property that

:<math>\widehat{\boldsymbol{u}}\boldsymbol{v} = \boldsymbol{u} \times \boldsymbol{v},</math>

where the left-hand side we have ordinary matrix multiplication. This implies <math>\boldsymbol{u}</math> is in the [[null space]] of the skew-symmetric matrix with which it is identified, because <math>\boldsymbol{u} \times \boldsymbol{u} = \boldsymbol{0}.</math>

===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}}

=== Isomorphism with 𝖘𝖚(2) ===
The Lie algebras <math>\mathfrak{so}(3)</math> and <math>\mathfrak{su}(2)</math> are isomorphic. One basis for <math>\mathfrak{su}(2)</math> is given by<ref>{{harvnb|Hall|2015}} Example 3.27</ref>

:<math>\boldsymbol{t}_1 = \frac{1}{2}\begin{bmatrix}0 & -i\\ -i & 0\end{bmatrix}, \quad \boldsymbol{t}_2 = \frac{1}{2} \begin{bmatrix}0 & -1\\ 1 & 0\end{bmatrix}, \quad \boldsymbol{t}_3 = \frac{1}{2}\begin{bmatrix}-i & 0\\ 0 & i\end{bmatrix}.</math>

These are related to the [[Pauli matrix|Pauli matrices]] by

:<math>\boldsymbol{t}_i \longleftrightarrow \frac{1}{2i} \sigma_i.</math>

The Pauli matrices abide by the physicists' convention for Lie algebras. In that convention, Lie algebra elements are multiplied by {{mvar|i}}, the exponential map (below) is defined with an extra factor of {{mvar|i}} in the exponent and the [[structure constant]]s remain the same, but the ''definition'' of them acquires a factor of {{mvar|i}}. Likewise, commutation relations acquire a factor of {{mvar|i}}. The commutation relations for the <math>\boldsymbol{t}_i</math> are

:<math>[\boldsymbol{t}_i, \boldsymbol{t}_j] = \varepsilon_{ijk}\boldsymbol{t}_k,</math>

where {{math|[[Levi-Civita symbol|''ε''<sub>''ijk''</sub>]]}} is the totally anti-symmetric symbol with {{math|1=''ε''<sub>123</sub> = 1}}. The isomorphism between <math>\mathfrak{so}(3)</math> and <math>\mathfrak{su}(2)</math> can be set up in several ways. For later convenience, <math>\mathfrak{so}(3)</math> and <math>\mathfrak{su}(2)</math> are identified by mapping

:<math>\boldsymbol{L}_x \longleftrightarrow \boldsymbol{t}_1, \quad \boldsymbol{L}_y \longleftrightarrow \boldsymbol{t}_2, \quad \boldsymbol{L}_z \longleftrightarrow \boldsymbol{t}_3,</math>

and extending by linearity.