Classical Lie groups

 Lecture topic

A Lie group $G$ is both a group and a manifold, with a smooth map $G\times G\to G$, given by $(g,h)\mapsto gh^{-1}$. The Lie algebra $\mathfrak g$ of $G$ is the tangent space $T_eG$ of $G$ at the identity.

We distinguish between real and complex Lie groups by saying that the base manifold is either real or complex analytic, respectively.

Example: Here are some examples of classical Lie groups and their dimension:
\[
\begin{array}{r c r c l}
\text{general linear group} & n^2 & GL(n) & = & \{n\times n\ \text{matrices with non-zero determinant}\} \\
\text{special linear group} & n^2-1 & SL(n) & = & \{M\in GL(n)\ :\ \det(M)=1\} \\
\text{orthogonal group} & n(n-1)/2 & O(n) & = & \{M\in GL(n)\ :\ MM^t = I\} \\
\text{special orthogonal group} & n(n-1)/2 & SO(n) & = & \{M\in O(n)\ :\ \det(M)=1\} \\
\text{unitary group} & n^2 & U(n) & = & \{M\in GL(n,\textbf{C})\ :\ MM^* = I\} \\
\text{special unitary group} & n^2-1 & SU(n) & = & \{M\in U(n)\ :\ \det(M)=1\} \\
\text{symplectic group} & n(2n+1) & Sp(n) & = & \{n\times n\ \text{matrices}:\ \omega(Mx,My)=\omega(x,y)\}
\end{array}
\]
For the symplectic group, the skew-symmetric bilinear form $\omega$ is defined as
\[
\omega(x,y) = \sum_{i=1}^n x_iy_{i+n} - y_ix_{i+n} = \begin{pmatrix} 0 & -I \\ I & 0 \end{pmatrix} x\cdot y,
\]
where $\cdot$ is the regular dot product (a symmetric bilinear form). Also note that the unitary group is a real Lie group - real because there is no holomorphic map $G\times G\to G$ as would be necessary, so we view the entries of a matrix in $U(n)$ in terms of its real and imaginary parts. Hence the dimension indicated above is real dimension.

References: Kirillov Jr (An introduction to Lie groups and Lie algebras, Chapter 2)