# Test for Lie Group

by mnb96
Tags: test
 P: 626 Hello, if I have a set of functions of the kind $\{ f_t | f_t:\mathbb{R}^2 \rightarrow \mathbb{R}^2 \; ,t\in \mathbb{R} \}$, where t is a real scalar parameter. The operation I consider is the composition of functions. What should I do in order to show that it forms a Lie Group?
 P: 1,411 First check if you have a group. Then check if the map $t\mapsto f_t$ is one to one. Next steps will depend on the results of these two.
 P: 626 Thanks arkajad, let's consider a toy example, in which we have that each ft is a linear function ℝ2→ℝ2 given by a counter-clockwise rotation of angle t around the origin, where t is in the range [0,2π). Each ft can effectively be represented as a 2x2 matrix, and the composition would be given by matrix multiplication. Matrix multiplication is associative, and for any pair of rotation matrices the composition yields another rotation matrix in the set, so we have closure too. Moreover we have an identity element which is f0, and also an inverse which is f2π-t ===> so, we have a group. We also consider the map $t \mapsto R(t)$, where R(t) is a 2x2 rotation matrix and t is the angle parameter. The map is clearly 1-to-1, provided t is in the range [0,2pi). Now, what else do we need?
 P: 1,411 In fact you want to give your group a parametrization. For this you do not have to consider orbits. You have your parametrization already: it is your t. Composition law in the group is effectively that of $t+t'$ modulo $2 \pi$. So, roughly speaking, all you need to do is checking that this composition law is differentiable (and also the inverse operation $t\mapsto 2\pi -t$). And addition of real numbers is differentiable in both variables. The only subtle point is "modulo $2 \pi$" part. It needs some mathematical precision, like covering the circle with two overlaping open segments - you can leave it aside for a while.
 P: 626 ok, thanks! so if my group is parametrized by say two parameters u, v, then I have to figure out how the group operation (composition of functions in our previous case) can be expressed in terms of the parameters u, v. In general, the composition will be $\phi:(U\times V)\times (U\times V) \rightarrow (U\times V)$, so basically I have to check that the 4-variable function: $$\phi(u_1,v_1,u_2,v_2)$$ is differentiable with respect to all the four variables. Am I right? If so, how am I supposed to do the same test for the inverse operation? Do I simply consider the function $$\phi(f(u_1),f(v_1),f(u_2),f(v_2))$$, where f denotes the inverse operation (in terms of parameters), and check that it is differentiable?