A Lie group $G$, named after the Norwegian mathematician Sophus Lie, is a group whose elements depend smoothly on some number of parameters, and on which the group operations \begin{align} G \times G &\longrightarrow G \\ (X,Y) &\longmapsto X^{-1}Y \end{align} are smooth. Most of the Lie groups considered here are matrix groups, whose elements are $n\times n$ matrices over some division algebra, and whose group operation is matrix multiplication. The simplest example is $\SO(2)$, the rotation group in two dimensions. As discussed in § 6.2, the elements of $\SO(2)$ take the form \begin{equation} M(\alpha) = \begin{pmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \\ \end{pmatrix} \end{equation} which clearly depends smoothly on $\alpha$. Furthermore, we have \begin{align} M(0) &= I \label{1pOne}\\ M(\alpha+\beta) &= M(\alpha)M(\beta) \label{1pAdd} \end{align} where $I$ denotes the (in this case $2\times2$) identity matrix. We refer to $\{M(\alpha):\alpha\in\RR\}$ as a 1-parameter family of group elements if it satisfies (\ref{1pOne}) and (\ref{1pAdd}). We are interested primarily in Lie groups that are connected to the identity, in which case every element belongs to at least one 1-parameter family.

The dimension $|G|$ of a Lie group $G$ is the number of independent parameters needed to describe it. For example, $\SO(2)$ is clearly a 1-dimensional Lie group. What about $\SO(3)$? There are three independent rotations in $\SO(3)$, which correclty suggests that the dimension of $\SO(3)$ is $3$. Alternatively, any rotation in three dimensions can be expressed as a product of three rotations about given, fixed axes, for instance using Euler angles. More generally, there are ${n\choose2}=\frac{n(n-1)}2$ independent planes in $n$ dimensions, so that \begin{equation} |\SO(n)| = \frac{n(n-1)}2 \end{equation} Similar arguments can be used to show that \begin{align} |\SU(n)| &= n^2-1 \\ |\Sp(n)| &= n(2n+1) \end{align}