I am reading a document about performing 2D and 3D transformations in space specifically rigid body transformations which can be represented using Lie groups. For example, the rigid body transformation in 2D can be represented by the SE(2) group which has two parameters for translation and one for rotation.
I do not have a very good mathematical background and I was reading a bit about the relationship between the lie group and the lie algebra and I read the following:
Every Lie group has an associated Lie algebra, which is the tangent
space around the identity element of the group
I am confused about what is so special about the tangent space at the identity element and as far as I know you are able to move from the tangent space i.e. the algebra to the group using an exponential map. So as I understand a vector in this tangent space will represent a particular velocity at the identity point and the exponential map will take us to the group element (in this case the identity).
So, I thought the algebra should consist of the set of tangent spaces at every element in the group and not just the identity element.