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\}

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