A Lie algebra, again named after Sophus Lie, is a vector space $\gg$ together with a binary operation \begin{align} \gg \times \gg &\longrightarrow \gg \\ (x,y) &\longmapsto [x,y] \end{align} called the Lie bracket of $x$ and $y$. The Lie bracket is bilinear and satisfies \begin{align} [x,y] + [y,x] &= 0 \label{Lie}\\ \quad[x,[y,z]] + [y,[z,x]] + [z,[x,y]] &= 0 \label{Jacobi} \end{align} where the second condition is known as the Jacobi identity. Lie algebras can be thought of as infinitesimal Lie groups. More formally, a Lie algebra is the tangent space of the Lie group at the origin, representing infinitesimal “displacements” in all possible “directions” there.

For matrix Lie groups, each 1-parameter family $M(\alpha)$ yields an element $A$ of the corresponding Lie algebra via \begin{equation} A = \dot{M} = \frac{\partial{M}}{\partial\alpha}\Bigg|_{\alpha=0} \label{Gtog} \end{equation} which is again a matrix. This operation can be reversed through matrix exponentiation (which can be defined formally using power series); we have \begin{equation} M(\alpha) = \exp(\dot{M}\alpha) \label{gtoG} \end{equation} The Lie bracket on a matrix Lie algebra is just the ordinary commutator, that is \begin{equation} [x,y] = xy - yx \end{equation} This operation on matrices clearly satisfies (\ref{Lie}), and a straightforward computation shows that (\ref{Jacobi}) is satisfied as well.

The simplest Lie algebra is $\so(2)$, the infinitesimal version of the Lie group $\SO(2)$, which is the vector space generated by \begin{equation} A = \frac{\partial}{\partial\alpha} \begin{pmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \\ \end{pmatrix} \Bigg|_{\alpha=0} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \\ \end{pmatrix} \end{equation} Thus, $\so(2)$ consists of all real multiplies of $A$, and is isomorphic to the 1-dimensional vector space $\RR$. All commutators on $\so(2)$ vanish.

The simplest nontrivial Lie algebra is $\so(3)$, the infinitesimal version of $\SO(3)$, which, as shown in § 7.3, is generated by ($-i$ times) the Pauli matrices. This factor of $i$ represents a major notational difference between two standard notations: Physicists typically insert an $i$ into the differentation operation (\ref{Gtog}), so that the resulting infinitesimal rotation matrices are Hermitian, whereas mathematicians typically omit this $i$, resulting in anti-Hermitian infinitesimal rotation matrices. Adopting this latter convention, $\so(3)$ is generated by \begin{align} \tau_x = -\frac{i\sigma_x}2 &= \frac12\begin{pmatrix} 0 & -i \\ -i & 0 \end{pmatrix}\\ \tau_y = -\frac{i\sigma_y}2 &= \frac12\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\\ \tau_z = -\frac{i\sigma_z}2 &= \frac12\begin{pmatrix} -i & 0 \\ 0 & i \end{pmatrix} \end{align} which satisfy the commutation relations \begin{align} \quad [\tau_x,\tau_y] &= \tau_z \\ \quad [\tau_y,\tau_z] &= \tau_x \\ \quad [\tau_z,\tau_x] &= \tau_y \end{align}

The dimension $|\gg|$ of a Lie algebra $\gg$ is its dimension as a vector space, which is the same as the dimension of the corresponding Lie group. Thus, $\so(3)$ is a 3-dimensional Lie algebra, and, more generally, \begin{equation} |\so(n)| = \frac{n(n-1)}2 \end{equation}