# Lie Groups Lecture 6

This lecture (lie6) was given on 9/15/09

### Excerpts

Given matrix lie groups and a matrix lie group homo., there exists a unique map that is basically a lie algebra homomorphism compatible with , meaning

here is how to construct : let , it can be checked that its a lie group whose lie algebra is a subset of , so any can be written as . Define via . Alternatively, using the result that every one parameter subgroup is we have is a one parameter subgoup hence , so define , the desire properties can be checked.

### Adjoint

A consequence of the lie product formula is that defined by is a lie group homomorphism, so by the above result there is such that . It can be checked, using 4 above, that .

The rest of the lecture we covered thm 2.27 through cor. 2.33 ( which says matrix lie groups are lie groups).

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- Published:
- September 16, 2009 / 10:42 pm

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- Lie Groups (course)

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