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I am trying to understand the difference between the "Drinfeld" and the "Lusztig" theory of quantum groups, more specifically with respect to the problem of quantization of Lie bialgebras/Poisson Lie groups.

There is a very well understood (and by now classical) theory of Etingof and Kazhdan on quantization of Lie bialgebras. For any (finite dimensional) Lie bialgebra $(\mathfrak{g}, \delta)$, one can construct a deformation Hopf algebra $U_h(\mathfrak{g})$ defined over $\mathbb{C}[[h]]$ quantizing $(\mathfrak{g}, \delta)$, and any quasitriangular $r$-matrix $r \in \mathfrak{g} \otimes \mathfrak{g}$ can be quantized to a quasitriangular $R$-matrix $R \in U_h(\mathfrak{g}) \hat{\otimes} U_h(\mathfrak{g})$.

My question is whether a similar quantization theory exists over $\mathbb{C}[q, q^{-1}]$. Can one always construct a Hopf algebra $U_q(\mathfrak{g})$ defined over $\mathbb{C}[q, q^{-1}]$ (or another similar looking polynomial ring) quantizing $(\mathfrak{g}, \delta)$ in a precise sense? Can any quasitriangular $r$-matrix be quantized to a universal $R$-matrix $R \in U_q(\mathfrak{g}) \hat{\otimes} U_q(\mathfrak{g})$?

My understanding is that the latter theory, if it exists would be better, as one would get an actual family of deformations, as opposed to a formal one.

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It all depends on what you mean by a well defined quantization theory.

In the case of standard semimsimple Lie bialgebra you can define elements $K_i=e^{\hbar d_i H_i}$ for each Cartan generator $H_i$, where $d_i$ are the natural numbers symmetrizing the Cartan matrix. Then you can show that letting $q=e^\hbar$ you have a $\mathbb C[q,q^{-1}]$ rational form $U_q(\mathfrak g)$ inside $U_\hbar(\mathfrak g)$. Finally, inside this rational form you still have an $R$-matrix of the form you considered.

All this is not functorial, however. In general you can ask for the existence of a suitable rational form inside the local deformation. Personally I have never seen an existence theorem of the form: inside each Etingof-Kazhdan $\hbar$-deformation there is a $q$-rational form. But I wouldn't be too much surprised if such a result holds true. Usually what is difficult is to identify one such rational form, since you cannot expect any uniqueness, and I guess that's why you rather go by generators and relations.

However, for the case of compact Poisson-Lie groups, since there you have a classification of Lie bialgebra structures, you can verify that any Lie bialgebra can be $q$-quantized and the $r$-matrix survives quantization. Best reference for this is the book by Korogodskii and Soibelman: Algebra of Functions on Quantum Groups I.

This discussion may be of help as well:

difference between $q$-deformations and $\hbar$-deformations

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  • $\begingroup$ Chari and Pressley also discuss integral/rational forms of QUE algebras in the chapter 9 of their book. $\endgroup$ – Victor Mouquin Jul 11 '18 at 2:29

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