Understanding definition of quantization of a Poisson-Hopf algebra

Question

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.

Answer 1

You didn't give the definition of $A_h$ but if you look there, you should see that elements of it are formal power series in the parameter $h$ with coefficients from $A$. Then "mod $h$" means "take the constant term of the power series, i.e. if $a=\sum a_i h^i$, $a \mod h=a_0$.

Then the condition you give means that if you look at two formal power series in the deformation, the commutation of their degree 1 coefficients with respect to $h$ is controlled by the Poisson bracket (of their degree parts). (Degree 1, from dividing through by $h$ and then taking mod $h$.)

In particular, if the Poisson bracket chosen is globally zero, then commuting of the degree 1 coefficients is undeformed.