# Lie Groups Lecture 1

This lecture took place on August 27th 2009.

### Formalities

Textbook: (Brian Hall) Lie Groups, Lie Algebras, and Representations.  Suggested online reading can be found here and there.  The first assingment is exercies 4,8,11 in chapter 1 of the text.

### Things I wrote down the first day

A Lie group is a smooth manifold G with a group structure such that multiplication and taking inverse are represented by smooth maps in the appropriate category.  In a mathematical situation, if there is a continuous symmetry, then there is like a Lie group lurking.  There is a powerful result (Montgomery-Zippin) that states manifold groups are Lie groups.

Some notation.  In general $\mathbb{K}$ will be a ring which in most cases will be $\mathbb{R,C,H}$.

Recall that the building blocks for finite groups are simple groups (Jordan Holder) in the sense that for any finite group G there is a normal series (descending chain of normal subgroups)

$G \rhd G_1 ... \rhd G_n \rhd 1$

Such that the successive quotients $G_i/G_{i+1}$ are simple groups.  The series is not unique but the quotients are.  We will see that Lie groups can be similarly decomposed.  For Lie groups the building blocks are simple/semisimple and solvable groups.

A matrix Lie group is a subgroups $G < GL(n, \mathbb{C})$ of the general linear group which is closed; i.e if $(A_m)_{m \in \mathbb{N}} \in G$ then for $A = \lim_{m \to \infty} A_m$ either $A \in G$ or $\det A = 0$.

### Examples

There is a correspondence between closed subgroups of $\mathbb{C}^*$ and closed subgroups of G s.t. $SL(n) < G< GL(n)$.  Namely, is $H \subset \mathbb{C}^*$ is closed then $\{A | \det A \in H\}$ is a closed subgroup.

(possibly temporary notation) $SL_{\pm}(n, \mathbb{R}) = \{A|\det A = \pm 1\}$ is the closed subgroup the preserves volume measure on $\mathbb{R}^n$.  $SL(n, \mathbb{R})$ preserves the volume form.

Note $GL(n, \mathbb{H}) < GL(2n, \mathbb{C})$ via

$\mathbb{H} = \left\{\left(\begin{array}{cc} a & b \\ -\bar b & \bar a \end{array} \right) \mbox{ s.t. } a, b \in \mathbb{C} \right\}$