# Lie Groups Lecture 4

This lecture ( lie4 ) was given on 9/8/09

### Excerpts

Let be matrix lie groups, a morphism between them is a continous group homomoprhism; if is a morphism then need not be a matrix lie subgroup. Consider the map defined by latex r\exp(i \alpha)$ for some irrational angle , the image is not a closed subgroup.

Polar decomposition for says any element in the special linear group can be written as a product where is unitary of determinant 1, and ; the set of positive Hermetian matrices of determinant 1

As an application, let . Then and are real and . Parametrize this as . This gives a vector space isomorphism between and . Also the det on gives minus the form associated to , namely . It can be check (using polar decomposition) that conjugation by an element of preserves , this gives a homomorphism with kernel being plus and minus the identity.

A lie group that is not a matrix lie group. acts on the n torus . It turns our for the semi direct product

is a lie group that is not a matrix lie group.

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- Published:
- September 9, 2009 / 10:18 am

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- Lie Groups (course)

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