# 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 will be a ring which in most cases will be .

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)

Such that the successive quotients 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 of the general linear group which is closed; i.e if then for either or .

### Examples

There is a correspondence between closed subgroups of and closed subgroups of G s.t. . Namely, is is closed then is a closed subgroup.

(possibly temporary notation) is the closed subgroup the preserves volume measure on . preserves the volume form.

Note via

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- August 28, 2009 / 10:32 am

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