For every finite dimensional semi-simple Lie group $\mathfrak{g}$, we have a loop algebra $\mathfrak{g}[t,t^{-1}]$. This loop algebra has a natural invariant inner product by taking the residue at zero of the Killing form applied to two elements (i.e. $t^k\mathfrak{g}$ and $t^{-k-1}\mathfrak{g}$ are paired by the Killing form.)

This Lie algebra actually has a Manin triple structure with respect to this inner product: the subalgebras $\mathfrak{g}[t]$ and $t^{-1}\mathfrak{g}[t^{-1}]$ are both isotropic, and non-degenerately paired by this form. This makes $\mathfrak{g}[t]$ into a Lie bialgebra, by getting the cobracket from the bracket on $t^{-1}\mathfrak{g}[t^{-1}]$.

Now, as we all know, Lie bialgebras can be quantized: in this case, the result is a quite popular Hopf algebra called the Yangian. By the usual yoga of quantization of Lie bialgebras, the dual Hopf algebra to the Yangian quantizes the universal enveloping $t^{-1}\mathfrak{g}[t^{-1}]$, so if you take a different associated graded of the Yangian, you must get the Hopf algebra of functions on the group with Lie algebra $t^{-1}\mathfrak{g}[t^{-1}]$, which is $L_<G$, the based formal loop group.

Now, all of these things also have explicit descriptions in terms of equations, and it seems as though this story must be worked out explicitly somewhere, but I've had little luck locating it. Does anyone know where? Or is this story just wrong, and that's why I can't find it?

EDIT: The comment below mostly answers this question. I would be interested if anyone out there has written something more explicit than the Etingof and Kazhdan paper, but it's the sort of thing I was looking for. If it were to be left in the form of an answer, I would probably accept it (hint, hint).