How to obtain a classical r-matrix from a quantum R-matrix?

Question

Let $R$ be a quantum R-matrix. Is there a procedure to dequantize $R$ and obtain a classical r-matrix? Thank you very much.

Answer 1

In the semiclassical (or quasiclassical) limit, by definition, the quantum $R$-matrix contains the classical $r$-matrix in the linear term in an expansion in a small parameter $\hbar$, so that $R \propto 1 + \hbar \, r + O(\hbar^2)$. Expanding the quantum Yang-Baxter equation and collecting terms linear in $\hbar$ yields the classical Yang-Baxter equation. (Of course in practice one should determine a suitable small parameter.)

It might be instructive to check the work by Belavin and Drinfeld. See e.g. the (more mathematical) book by Chari and Pressley or the (more physics) book by Korepin, Bogoliubov and Izergin.

Edit: Let me add that the terminology "semiclassical", like quite a lot of terminology in this field, comes from quantum mechanics. See e.g. this nLab entry, or this phyiscs stackexchange answer.