Lie groups

20 janeiro 2022, 12:30 • John Huerta

We completed our differential geometry toolkit with the Lie bracket of vector fields: given any two vector fields X and Y are a manifold M, the Lie bracket [X,Y] on M is the operator on smooth functions defined by taking the commutator of X and Y:

Prop. [X,Y] is a vector field on M.

Using coordinates for the proof makes essential use of the fact that partial derivatives commute.

We then defined a Lie group: a Lie group G is a group which is also a manifold such that the group operations are smooth. We stated Cartan's theorem on closed subgroups, and finished up by giving several examples of Lie groups:

1. (ℝⁿ, +) is an abelian Lie group.
2. The real general linear group GL_n(ℝ) is, first of all, an open set in the set of n x n real matrices, Mat_n(ℝ). This makes it a manifold with one chart, and it is easy to check the group operations, given by matrix multiplication and matrix inversion, are both smooth. Hence, GL_n(ℝ) is a Lie group.
3. Similarly, the complex general linear group GL_n(ℂ) is a Lie group.
4. (a) The real special linear group SL_n(ℝ) is the subgroup of GL_n(ℝ) consisting of matrices with determinant 1. Because the determinant is a continous function, SL_n(ℝ) is a closed subgroup. Hence, SL_n(ℝ) is a Lie group by Cartan's theorem. Similarly, the complex special linear group SL_n(ℂ) is a Lie group.
   

• (b) The real orthogonal group O_n(ℝ) is the subgroup of GL_n(ℝ) consisting of matrices A that satisfy A^TA = I. Because the map sending A to A^TA is continuous, this is a closed subgroup. Hence, O_n(ℝ) is a Lie group by Cartan's theorem. Similarly, the complex orthogonal group O_n(ℂ) is a Lie group.