# Is there a good reference for the relationship between the Yangian and formal based loop group?

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).

• Thanks for the hint. I changed my comment to an answer. Mar 22, 2010 at 9:47
• Well, I might misunderstand the question, but it seems to me everything is explicit, and more or less isomorphic to each other. There is "Yangian double" arxiv.org/abs/hep-th/9406194 arxiv.org/abs/q-alg/9602031 which contains both positive and negative parts of loop group quantization and all parts are explicitly described. It is the same story as for quantized loop group U_q(gl((t)) - there is no conceptual difference. As in non quantum case gl[t] gl[t^-1] - are isomorphic, more or less the same after quantization Apr 5, 2020 at 9:43