Generator of Time Shift in Classical and Quantum Mechanics The time evolution of a point in phase space in classical mechanics can be described as
\begin{equation}\label{eq:TmeShift}
 ( q_i(t + \Delta t),p_i(t + \Delta t) ) = \left( 1 - i\Delta t \hat{L}\right)   (q_i(t),p_i(t) ),
\end{equation}
where $\Delta t$ - infinitesimal time shift, $\hat{L}$ - Liuouvillian, given as
\begin{equation}\label{eq:Liuvillian}
 \hat{L} = i \sum_{i=1}^n \left[\frac{\partial H}{\partial p_i}\frac{\partial}{\partial q_i}-\frac{\partial H}{\partial q_i}\frac{\partial }{\partial p_i}\right]=i\{\cdot,H\},
\end{equation}
So, I can conclude in accord with Lie group theory that $\hat{L}$ is the generator of time shifts. I am a little bit confused why books says that the classical Hamiltonian is the generator of time shift in classical mechanics, the same as in QM, rather than the Liuovillian. Why is that so?
 A: You are correct in saying that the time shift is generated by $\{\cdot,H\}$, not by $H$ itself.
In fact, for any classical quantity $A(p,q)$ we can construct a classical operator (vector field) in phase space as
$$\hat{A}_{\rm CM} \equiv \{\cdot,A(p,q)\} = \sum_i \frac{\partial A}{\partial p_i} \frac{\partial}{\partial q^i} - \frac{\partial A}{\partial q^i} \frac{\partial}{\partial p_i}$$
The quantum operator $\hat{A}_{\rm QM}$ is then sometimes easier to understand as corresponding to $\hat{A}_{\rm CM}$ rather than $A$ itself. For instance, the commutators correspond in a straightforward manner
$$[\hat{A}_{\rm CM},\hat{B}_{\rm CM}] \sim i [\hat{A}_{\rm QM},\hat{B}_{\rm QM}]$$
That is, one should understand the statements "this and this transform is generated by $A(p,q)$" as, in fact, "this and this transform is generated by $\hat{A}_{\rm CM}$". In this sense all transforms are generated by the same operators in classical and quantum mechanics (and in some constructions of QM this is the basis of their correspondence).
