Let $SO(3)$ be the rotation groups with its natural action in $\mathbb R^3$: $$\vartheta:(A,x)\in SO(3)\times \mathbb R^3\longmapsto Ax\in\mathbb R^3$$ I was asking myselft if this action induce a traslation action: $$\mathcal t: (u,x)\in\mathbb R^3\times\mathbb R^3\longmapsto u+x$$ The idea is that the lie algebra of $SO(3)$ can be view as vectos in $\mathbb R^3$ with the cross product. But I cannot give explicity the translation action $t$ in terms of $\vartheta$. Is this possible?
Thanks in advance.
No. The translation group $\mathbb{R}^3$ and the rotation group $SO(3)$ both acts on the same space ($\mathbb{R}^3$) but they are really different. You can form a big group, sometimes called $E(3)$, the Euclidean group, consisting of all translations and rotations and their products, that also acts on $\mathbb{R}^3$.
However you can't get a translation as a product of rotations, as is easy to see since all rotations fix the origin. And also not as an infinitesimal linear approximation to a rotation (i.e. as an element of the Lie algebra) since it would translate different elements in different directions rather than in the same direction.
If we interpret elements of the Lie-algebra as vectors rather than movements of space, then the direction of this vector corresponds to the rotation axis of the corresponding rotation, so perpendicular to the movement.
(Edit, I just remembered: the standard definition of the Eucledian group $E(3)$ also contains reflections and this subgroup of orientation preserving elements has a different name: $E_+(3)$ or something like this. You should check Wikipedia)