Suppose I have a Lie group action $$ G\times M\to M, (g,m)\mapsto g\cdot m. $$ which is transitive on $M$, then the tangent functor $T$ induces a corresponding map: $$ TG\times TM\to TM, (\delta g,\delta m)\mapsto \delta g\cdot m+g\cdot\delta m. $$ Note that given the group operation of $G$: $$ G\times G\to G, (g,h)\mapsto g\cdot h. $$ $TG$ can be induced a Lie group structure by: $$ \delta g\star\delta h:=\delta (g\cdot h)=\delta g\cdot h+g\cdot\delta h. $$ The group axioms can be verified as follows:
- closure: trivial;
- identity: $0_e$ where $e$ is the identity of $G$;
- inverse: $(\delta g)^{-1}=\delta(g^{-1})$;
- associativity: $$ (\delta g\star\delta h)\star\delta k=\delta(g\cdot h)\star\delta k=\delta (g\cdot h\cdot k)=\delta g\star\delta(h\cdot k)=\delta g\star(\delta h\star\delta k). $$ Under this particular group operation of $TG$, the induced map $TG\times TM\to TM$ defines a Lie group action of $TG$ on $TM$, which is evident from Leibniz rule. And in particular it should be transitive too.
Questions:
- I would like someone to verify for me my above claims first.
- If there is a good reference about it, I would very much like to know.
- Is it possible to define an induced action $TSO(3)\times TS^2\to TS^2$ from: $$ SO(3)\times S^2\to S^2, (R,r)\mapsto Rr $$ such that it corresponds exactly to the Adjoint action $SE(3)\times se(3)\to se(3)$? (see the comments below)
Some follow-up comments:
One of the main applications is a generalization of the action $SO(3)\times S^2\to S^2$ (with $SO(3)$ considered as a Lie subgroup of $SE(3)$) to the whole of $SE(3)\simeq TSO(3)$ on $TS^2$:
$$
\begin{split}
\psi:SE(3)\times TS^2&\to TS^2, \\
(\hat pR,\delta r)&\mapsto \hat pRr+R\delta r.
\end{split}
$$
where $p\in\mathbb R^3, R\in SO(3), r\in S^2$ and $\wedge$ is defined by:
$$
\wedge:(x,y,z)^T\mapsto\left[\begin{array}{ccc}
0 & -z & y\\
z & 0 & -x\\
-y & x & 0
\end{array}\right]
$$
and investigate its connection to Adjoint transformation:
$$
Ad:SE(3)\to\mathfrak{gl}(se(3))
$$
To see the connection, define a $\xi\in se(3)$ for $\delta r\in TS^2$ by:
$$
\xi:=\left[\begin{array}{cc}\hat r & r\times\delta r\\
0 & 0
\end{array}\right], r\in S^2.
$$
Unfortunately, the induced action $TSO(3)\times TS^2\to TS^2$ is not consistent with the Adjoint action. I am looking for an alternative action $TSO(3)\times TS^2\to TS^2$ such that it is.
The first 2 questions are more or less confirmed by comments. The 3rd question is easily solved by defining the twist $\xi$ as: $$\left[\begin{array}{cc} \hat r & \delta r\\0 & 0\end{array}\right]$$ Then the induced action becomes coincident with the Adjoint action. should have thought about that...