Lie Groups

Lie Groups of Finite Dimension

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 .

Matrix Lie Groups

Matrix Lie Group

A matrix Lie group is a closed subgroup of , using the standard topology from .

Attention

We have not yet established that a matrix Lie group is indeed a Lie group. This will be proved later, after introducing the exponential map for complex matrices.

e.g. In fact, most of common Lie groups, and Lie groups that are used in physics are matrix Lie groups:

• Orthogonal Group: , .
• Unitary groups: , .
• Symplectic group: .

But clearly not every Lie group is a matrix Lie group. Here are two examples of non-matrix Lie groups:

• Elliptic curves over .
• The universal cover of .

Compactness

One benefit of matrix Lie groups is that it is easy to check compactness by Heine-Borel theorem. For example, and are compact since every orthogonal/unitary matrix has Frobenius norm , and so is bounded. On the other hand, is not compact since the subgroup is not bounded.

Connectedness

Checking connectedness is not hard as well. , , , , are all connected matrix Lie groups. Besides, we have the following result:

Proposition

Let be a matrix Lie group, and let be the path connected component of the identity . Then is a normal subgroup of .

Proof We first show that is a subgroup. For , there is a path joining and . Suppose is the left multiplication by , and is the inversion map. Then is a path joining and , hence , so is a subgroup. For any , consider the conjugation map defined by . Since is continuous and , joins and , hence . Therefore, is a normal subgroup of .