In Parametrization of SL(2,R) we introduced global coordinates on the group SL(2,R). Any matrix in SL(2,R) can be uniquely written as
If is the matrix with components then its coordinates can be expressed as functions of the matrix components as follows
The function returns the angle , of the complex number
Once we have coordinates, it is easy to calculate components of tangent vectors to any given path – they are given by derivatives of the coordinates We will calculate now the vector fields resulting from left and right actions of one-parameter subgroups of SL(2,R) that we have already met in SL(2,R) generators and vector fields on the half-plane
We have introduced there the one-parameter groups, that we will denote now as , generated by :
We can take exponentials of the generators and construct one-parameter subgroups
In SL(2,R) generators and vector fields on the half-plane we were acting with these transformations on the upper half-plane using fractional linear representation. Now we will be acting on SL(2,R) itself via group multiplication, either left or right.
Let us start with acting from the left. At we have the trajectory . Its coordinates are Let us denote by the tangent vector field, with components Then
We can calculate it easily using algebra software. The result is:
The same way we get
Then we can calculate vector fields of right shifts
We know that our metric is bi-invariant. That means the vector fields of the left and right shifts generate one-parameter group of isometries. They are called Killing fields of the metric. In differential geometry one shows that a vector field is a Killing vector field for metric if and only if the Lie derivative of the metric vanishes. Lie derivative of any symmetric tensor is defined as
Using any computer algebra software it is easy to verify that Lie derivatives of our metric with respect to all six vector fields indeed vanish. Mathematica notebook verifying this property can be downloaded here.