I'm having a little difficulty wrapping my head around Lie theory (I'm a computer scientist, so perhaps that's to be expected).
Specifically, considering the following definition from Wikipedia for the exponential map. What is the significance of the identity element? I think that this could be to do with the derivative of exp being equal to the function itself, but I'm not quite sure if I'm on the right track here. Any nudges in the right direction would be highly appreciated.
"Let G be a Lie group and $\mathfrak{g}$ be its Lie algebra (thought of as the tangent space to the identity element of G). The exponential map is a map $\exp\colon \mathfrak g \to G$ which can be defined in several different ways."
Best Answer
The answers given are both really good; I just want to add an additional reason why the identity element is preferred in consideration of the exponential map. The reason is because the exponential map gives a one to one correspondence between tangent vectors at the identity and one parameter subgroups of $G$. A one parameter subgroup of $G$ is a homomorphism $\mathbb R \to G$, or equivalently a curve $\gamma(t)\in G$ such that $\gamma(t+s) = \gamma(t)\gamma(s)$. Since $\gamma$ is a group homomorphism, we need $\gamma(0)$ to be the identity $e\in G$ (so here we are using the special property of $e$) and any such one parameter subgroup is determined by $\gamma'(0) \in T_e G$. Conversely, given $X \in T_e G$, the curve $t\mapsto \exp(tX)$ is a one-parameter subgroup with derivative $X$ at time 0.