Slutsky Decomposition from Indirect Utility Function Given the indirect utility function:
V={M^2}/{4P1P2},
how do we establish the Slutsky Decomposition?
I used Roy's Identity to get the Demand, but I'm stuck with the other components of the Slutsky Equation.
Since you already know the demand correspondences, you can use use the duality principle as follows: $v(\textbf{p},e(\textbf{p},u)) = u.$ Using this we get $\frac{4e(\textbf{p},u)}{p_1p_2}=u$, or $e(\textbf{p},u)) = \frac{up_1p_2}{4}.$ Now, using Shephard's Lemma, we can obtain the Hicksian demands as: $\frac{\partial e(\textbf{p},u)}{\partial p_i} = h_i(\textbf{p},u)).$ 

However, the job is not yet done. You might get stuck with the $u$ while evaluating the Slutsky's equation. In other words, you need to find the utility function as well. The demand correspondences are(I hope you've obtained the same expressions) $x_1(p_1,p_2,m)=m/2p_1$ and $x_2(p_1,p_2,m)=m/2p_2$. Since the indirect utility function is homogenous of degree zero, you can further write it as $v(p_1,1,m)=v(p,m)=m^2/4p=(m/2).(m/2p).$ From the demand correspondences obtained, you have to represent $m/2$ and $m/2p$ in terms of $x_1$ and $x_2$ ($p_2=1$, as a consequence of homogeneity of $v(p_1,p_2,m)$), and substitue it back into the indirect utility function. Thus, we obtain the the utility function $U(x_1,x_2)=x_1.x_2$(since we obtain the indirect utility function by substituting the walrasian demands into the given utility function, backtracking the same steps gives us back the utility from the indirect utility). I think now one can easily evaluate the Slutsky's Equation.