Lie Groups

Lie Group

A Lie group is a group that is also a smooth manifold, such that the group operations of multiplication and inversion are both smooth maps. That is is a smooth mapping of the product manifold into .

e.g. The group of rotations in , , is a Lie group. Each element of can be represented by some rotation angle , this gives a chart so that is a smooth manifold. And the multiplication operation corresponds to the addition of angles, which is smooth. In general, are all Lie groups.

Proposition

Let be a Lie group. Then the left multiplication by , , is a diffeomorphism of onto itself.

Proposition

For any Lie group of dimension , we have with respect to a basis for .

Proof Define a map by , which is linear along each fiber (tangent space) of .

Adjoint Representation (Action)