# Lie Groups lecture 2

This lecture was given on september 1st 2009.

This lecture concerned important examples of matrix lie groups: $GL(n, k), SL(n,k), O(n), SO(n), U(n), SU(n), SP(n), O(n,m), SO(n,m)$.  Most of the material is taken from chapter 1 of Brian Hall’s book.  Also a classification of bilinear forms was presented.

The notation is that $O(B)$ is the group of invertible matrices that preserve $B$: $B(Ax,Ay) = B(x,y)$.  In general, it suffices to study symmetric and anti symmetric bilinear forms because it can be written as a sum of symmetric and antisymmetric forms

$B = \frac{B + B^\intercal}{2} + \frac{B - B^\intercal}{2}=: B' + B''$

$O(B) = O(B') \cap O(B'')$

### Symmetric Bilinear Forms

Assume $B$ is symmetric.  Then its normal so the spectral thm says there exists a basis under which its orthogonal.  Its also Hermitian so its eigenvalues are real.  So after scaling can assume $B = I_p\oplus -I_n \oplus 0_m$ where $p + n + m = N$ is the dimension of the vector space.  Let this basis be given by $v_1, ...., v_N$.  Define

$m = \dim V_0, V_0 = \{x \in \mathbb{R}^N : B(x,\mathbb{R}^N) = 0 \}$

$V_\pm = span \{v_i : \pm B(v_i, v_i) > 0\}$

The inertial thm says if $W: w_1, ...., w_N$ is another basis with $W_0 = V_0$ then $\dim W_\pm = \dim V_\pm$.  If not, say $\dim W_+ > \dim V_+$ then $\dim W_+ + \dim(V_0 + V_-) > N$ so there subspaces have to intersect nontrivially giving a $v$ such that $B(v,v) > 0$ and $B(v,v) \le 0$; gives contradiction.

### Anti Symmetric Bilinear forms

If $B$ is antisymmetric ($B(x,y) = -B(y,x)$ ) then define $V_0 \subset k^n$ as $B(V_0, k^n) = 0$.  Then $k^n$ can be decomposted $k^n = V_0\oplus W$.  By retricting to $W$ can assume $B$ is nondegernate.  Choose $x \in W$ arbirary, by assumption there exists $y$ s.t. $B(x,y) \ne 0$.  Let $V_1 = span(x,y)$ and decompose $W = V_1 \oplus V_1^{\perp}$ where $V_1^{\perp} = \{ w | B(w,V_1) = 0\}$.  Repeating the argument (and noting that B is nondegenerate) it follows that $W = V_1 \oplus ... \oplus V_m$ where each $V_i$ are two dimensional and (afer scaling) $B|_{V_i} = \left(\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix}\right) =: H$.  It follows that $W$ is even dimensional and the contruction gives a matrix for $B$ which is a direct sum of $H$‘s.

Its a fac that $B$ also has a matrix of the form $J:= \left(\begin{smallmatrix} 0 & I \\ -I & 0 \end{smallmatrix}\right)$.

It can be checked that $GL(n, \mathbb{C}) \subset GL(2n, \mathbb{R})$ by identitifying it with the $\{ A : AJ = JA\}$.  If $\mathbb{C}^n \ni x,y = (x_1, ...., x_n), (y_1, ..., y_n)$ are decomposed as $x_k = u_k + iv_k$ and $y_k =w_k +iz_k$ taking the usual norm on $\mathbb{C}^n$ gives

$\langle x,y \rangle = \sum_k u_kw_k + v_kz_k + i\sum_k u_kz_k - v_kw_k$

Which separate it into a symmetric and antisymmetric part.  Consequently $U(n) = \{A: AJ = JA\} \cap O(2n, \mathbb{R}) \cap SP(n, \mathbb{R}) \subset GL(2n, \mathbb{R})$.