# Lie Groups Lecture 7

This lectue (lie7) was given on 9/17/09.

### Excerpts

Recall takes . Now for every ,

Also, . It follows iff

So the Jacobi identity is equivalent to the adjoint representation being a homomorphism of lie algebras.

Some results: (thm 2.40; Ado) Every finite dimensional lie algebra is isomorphic to a subalgebra of for some n (similar result for ).

Given a basis for a lie algebra you can determine the structure of the lie bracket by the structure constants .

A real lie algebra can be complexfied and the lie bracket extends uniquely .

Some of the material in Appendix C was covered. Let be a smooth manifold with a group structure. The tangent space at a point is

You also require a condition like if in a nbd of , then . Left multiplication gives a map . Fixing a vector a left invariant vector field can be defined via . This gives a vector space isomorphism between and the set of left invariant vector fields. Also, there is a notion of taking the bracket of vector fields. A vector field is derivation , hence they can be composed, so it makes sense to take of two vector fields.

In this way we can define a Lie bracket on by thinking of vectors as left invariant vector fields and taking the bracket as above.

Alternatively, for every , there is an adjoint map . Recall . You can take the differential at the identity and get a representation . Taking differential of this you get , then can define .

Some questions

- Is the description as left invariant vector fields consistent with the definition using adjoint representation
- What happens when you define things with right invariant vector fields
- What is the bracket of a left invaraint vector fields with a right invariant vector field.

The lower central series is the unbalanced one: ; the derived series is the symmetric one . Nilpotent means the lower central series terminates, solvable means the derived series terminates and nilpotent implies solvable.

A lie algebra is simple is it has no notrivial ideals; semisimple means no notrivial solvable ideals.

## About this entry

You’re currently reading “Lie Groups Lecture 7,” an entry on Math Meandering

- Published:
- September 20, 2009 / 4:30 pm

- Category:
- Lie Groups (course)

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