I am going through the chapter *Quantization of Lie bialgebras* from the book *A Guide to Quantum Groups* by Chari and Pressley. There I found a notion called *Quantization* which deals with deformations of Hopf algebras of certain kind. To begin with, the authors first described what is called a quantization of a commutative *Poisson-Hopf* algebra $A.$ As I said, first of all it is a deformation $A_h$ of $A$ as a Hopf algebra. Secondly, it has to satisfy the following condition $:$

$$\{x_1, x_2 \} \equiv \frac {a_1 a_2 - a_2 a_1} {h}\ (\text {mod}\ h),$$

if $a_1, a_2 \in A_h$ reduce to $x_1, x_2 \in A\ (\text {mod}\ h).$

The second condition is something which I am unable follow properly. First of all if we are begin with a commutative Poisson-Hopf algebra then doesn't that imply that the RHS of that modularity condition is zero or am I misinterpreting of what is actually being meant to say? Secondly, what do the authors mean by the phrase "... if $a_1, a_2 \in A_h$ reduce to $x_1, x_2 \in A\ (\text {mod}\ h)$"? Could anyone please give me some insight on what is happening here? Any help in this regard would be warmly appreciated.

Thanks for your time.