# Understanding definition of quantization of a Poisson-Hopf algebra

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.

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.

• But if $A$ is commutative so is $A_h.$ Right? Then what about the commutator $[a_1, a_2],$ for $a_1, a_2 \in A_h\$? Will it not be zero and so is it's coefficient of degree $1$ term? If yes, then that will imply $A$ has a trivial Poisson structure. Right? But I still think I am misinterpreting something which is actually being meant to say by the two resourceful authors. I am wondering on where did I mess things up. It will be great if you please clarify it to me. Thanks a lot for your help. Sep 14 at 7:03
• Taking $A$ and the zero Poisson bracket, $A_h$ is just the ("undeformed") formal power series algebra over $A$, $A[[h]]$. That is true no matter what properties $A$ has. If $A$ is commutative and the Poisson bracket is non-zero, you will typically get a noncommutative algebra (i.e. a deformation). Does that help? Sep 16 at 7:43