Let $G$ be a Lie group. Recall the Lie logarithm is well-defined about a neighborhood $U \subset G$ of the identity: $\log:U\to \mathfrak{g}$. I am dealing with a research problem that concerns the differential of the Lie logarithm on $U$ $d\log_x: T_xG \to T_{\log(x)} \mathfrak{g}$, where $x \in U$. In particular, I am dealing with $G=SO(3)$, but I am trying to be as general as possible.
Recall that $d\log_x$ has a closed-form expression when we identify both $T_xG$ and $T_{\log(x)}\mathfrak{g}$ with $\mathfrak{g}$. If we denote the resulting operator as $A_x:\mathfrak{g} \to \mathfrak{g}$, then $$A_x(\xi)=\sum_{k=0}^\infty \frac{(-1)^k B_k}{k!}\text{ad}_{\log(x)}^k(\xi),$$ where $\xi \in \mathfrak{g}$ and $B_k$ is the $k$th Bernoulli number.
I'm very curious about the linear analysis of $A$. I am aware that $(1,\log(x))$ is an eigenvalue-eigenvector pair. I am also aware this operator is invertible for some neighborhood (everywhere?) about $e$. But that is really it.
I have some questions:
- Is there any way we can completely characterize the eigenvalues and eigenvectors of $A_x$?
- Does $A_x$ admit an eigenbasis?
- Is $A_x$ always invertible for any $x$ such that $\log(x)$ exists?
- If we equip $G$ with a bi-invariant metric (assuming one exists), then is there a relationship between $d(x,e)$ and how far $A_x$ is from $id:\mathfrak{g} \to \mathfrak{g}$? For distance on Lie algebra linear operators, I'm thinking of simply the operator norm induced from the Riemannian norm on $\mathfrak{g}$. Clearly $x \to e$ implies $A_x \to id$.
- Are there any papers or textbooks that dive into an analysis of the differential of the Lie logarithm?
