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 is denoted by \mathfrak{so}(3) and consists of all
skew-symmetric matrices. This may be seen by differentiating the
orthogonality condition, . The Lie bracket of two elements of \mathfrak{so}(3) is, as for the Lie algebra of every matrix group, given by the matrix
commutator, , 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 \mathfrak{so}(3) 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 R(\phi, \boldsymbol{n}) denotes a counterclockwise rotation with angle \phi about the axis specified by the unit vector \boldsymbol{n}, then :\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}. This can be used to show that the Lie algebra \mathfrak{so}(3) (with commutator) is isomorphic to the Lie algebra \R^3 (with
cross product). Under this isomorphism, an
Euler vector \boldsymbol{\omega}\in\R^3 corresponds to the linear map \widetilde{\boldsymbol{\omega}} defined by \widetilde{\boldsymbol{\omega}}(\boldsymbol{u}) = \boldsymbol{\omega}\times\boldsymbol{u}. In more detail, most often a suitable basis for \mathfrak{so}(3) as a vector space is : \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}. The
commutation relations of these basis elements are, : [\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 which agree with the relations of the three
standard unit vectors of \R^3 under the cross product. As announced above, one can identify any matrix in this Lie algebra with an Euler vector \boldsymbol{\omega} = (x,y,z) \in \R^3, :\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). This identification is sometimes called the
hat-map. Under this identification, the \mathfrak{so}(3) bracket corresponds in \R^3 to the
cross product, :\left [\widehat{\boldsymbol{u}},\widehat{\boldsymbol{v}} \right ] = \widehat{\boldsymbol{u} \times \boldsymbol{v}}. The matrix identified with a vector \boldsymbol{u} has the property that :\widehat{\boldsymbol{u}}\boldsymbol{v} = \boldsymbol{u} \times \boldsymbol{v}, where the left-hand side we have ordinary matrix multiplication. This implies \boldsymbol{u} is in the
null space of the skew-symmetric matrix with which it is identified, because \boldsymbol{u} \times \boldsymbol{u} = \boldsymbol{0}.
A note on Lie algebras In
Lie algebra representations, 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, \boldsymbol{J}_x, \boldsymbol{J}_y, \boldsymbol{J}_z, of the algebra : [\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. That is, the Casimir invariant is given by :\boldsymbol{J}^2\equiv \boldsymbol{J}\cdot \boldsymbol{J} =\boldsymbol{J}_x^2+\boldsymbol{J}_y^2+\boldsymbol{J}_z^2 \propto \boldsymbol{I}. For unitary irreducible
representations , the eigenvalues of this invariant are real and discrete, and characterize each representation, which is finite dimensional, of dimensionality 2j+1. That is, the eigenvalues of this Casimir operator are :\boldsymbol{J}^2=- j(j+1) \boldsymbol{I}_{2j+1}, where is integer or half-integer, and referred to as the
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
doublet (
spin-1/2) representation. By taking
Kronecker products of with itself repeatedly, one may construct all higher irreducible representations . That is, the resulting generators for higher spin systems in three spatial dimensions, for arbitrarily large , can be calculated using these
spin operators and
ladder operators. For every unitary irreducible representations there is an equivalent one, . 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 characterize
bosonic representations, while half-integer values
fermionic representations. The
antihermitian matrices used above are utilized as
spin operators, after they are multiplied by , so they are now
hermitian (like the Pauli matrices). Thus, in this language, : [\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. and hence :\boldsymbol{J}^2= j(j+1) \boldsymbol{I}_{2j+1}. Explicit expressions for these are, :\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} where is arbitrary and 1 \le a, b \le 2j+1. For example, the resulting spin matrices for spin 1 (j = 1) are :\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} Note, however, how these are in an equivalent, but different basis, the
spherical basis, than the above
L in the Cartesian basis. For higher spins, such as spin (j=\tfrac{3}{2}): :\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} For spin (j = \tfrac{5}{2}), :\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}
Isomorphism with 𝖘𝖚(2) The Lie algebras \mathfrak{so}(3) and \mathfrak{su}(2) are isomorphic. One basis for \mathfrak{su}(2) is given by :\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}. These are related to the
Pauli matrices by :\boldsymbol{t}_i \longleftrightarrow \frac{1}{2i} \sigma_i. The Pauli matrices abide by the physicists' convention for Lie algebras. In that convention, Lie algebra elements are multiplied by , the exponential map (below) is defined with an extra factor of in the exponent and the
structure constants remain the same, but the
definition of them acquires a factor of . Likewise, commutation relations acquire a factor of . The commutation relations for the \boldsymbol{t}_i are :[\boldsymbol{t}_i, \boldsymbol{t}_j] = \varepsilon_{ijk}\boldsymbol{t}_k, where is the totally anti-symmetric symbol with . The isomorphism between \mathfrak{so}(3) and \mathfrak{su}(2) can be set up in several ways. For later convenience, \mathfrak{so}(3) and \mathfrak{su}(2) are identified by mapping :\boldsymbol{L}_x \longleftrightarrow \boldsymbol{t}_1, \quad \boldsymbol{L}_y \longleftrightarrow \boldsymbol{t}_2, \quad \boldsymbol{L}_z \longleftrightarrow \boldsymbol{t}_3, and extending by linearity. ==Exponential map==