Rotation of $su(2)$ generators

Question

I am confused by conjugation, and the action of group elements on themselves. If I have a rotation acting on the generators of $su(2)$, i.e.

\begin{align} R_\theta (L_1, L_2, L_3) \end{align}

where $R_\theta$ is some rotation matrix. The elements of the vector $(L_1, L_2, L_3)$ are themselves the $su(2)$ generators, in an $N$-dimensional representation. Is it true that this action can be equivalently stated as a conjugation of the elements themselves? i.e.

\begin{align} R_\theta (L_1, L_2, L_3) \leftrightarrow (\Omega L_1 \Omega^\dagger, \Omega L_2 \Omega^\dagger, \Omega L_3 \Omega^\dagger) \end{align}

where $\Omega$ is defined by, for the example of a rotation around the $z$-axis,

$$\Omega = e^{i\theta L_3}$$

Answer 1

I think that you are groping for the equation $$ U(R) \sigma_i U^{-1}(R)= \sigma_j {R^j}_i, $$ where ${R^j}_i$ is an ${\rm SO}(3)$ rotation matrix, the $\sigma_i$ are the matrix generators of $\mathfrak{su}(2)$ in some representation, and the $U(R)$ are the corresponding ${\rm SU}(2)$ matrices in the same representation.

This is saying that the the adjoint action of the group ${\rm SU}(2)$ on its Lie algebra implements the ${\rm SO}(3)$ rotation group.