# Drinfeld Polynomial for Yangian $Y(\mathfrak{sl}_2)$

I am looking for a direct proof that a highest weight representation of $$Y(\mathfrak{sl}_2)$$ is finite-dimensional if its highest weight is determined by a Drinfeld polynomial.

The results was figured out in Tarasov's paper in 1984 and also was claimed in Drinfeld's paper in 1988.I found a proof of the similar result for the quantum affine algebra $$U_q(\widehat{\mathfrak{sl}}_2)$$ in the parper by Chari and Pressley in 1991 [Quantum Affine Algebras, Comm. Math. Phys. 142]. My question is: Is it possible to formulate a proof similar to Chari and Pressley for Yangian $$Y(\mathfrak{sl}_2)$$.

I understand that there is detailed proof in Molev's book. But that proof used the fact that a finite irreducible highest weight representation is a tensor product of evaluation representations. I wish to find a proof based on a direct calculation.

Any comments and suggestions are appreciated.

• Can't you adapt Chari and Presley's proof as $q \to 1$? (I don't have it here, but maybe they happen to do this in their book "A guide to quantum groups"?) – Jules Lamers Jun 26 at 16:56
• @JulesLamers, I tried but did not succeed. In Chari and Presley's book, only quantum affine case was proved. It seems that the results for Yangians are naturally transferred from those for quantum affine algebras. But I do not understand how it works. Any reference recommended? Thank you! – Zhihua Chang Jun 27 at 2:03
• Maybe you already know all of this better than I do, but just in case: you can get the rational (xxx) R-matrix, which through the 'RTT relations' gives rise to the defining relations of the Yangian (after rearranging the terms), from the trigonometric (six-vertex/xxz) R-matrix, which does likewise in the quantum affine case, by writing $q=e^{\hbar}$ and (from the top of my head, for multiplicative spectral parameter) $u=e^{\hbar \lambda}$, and looking at terms linear in $\hbar$. – Jules Lamers Jun 27 at 20:13
• (cont'd) If you understand this limit at the level of the R-matrix you should be able to do likewise for the defining relations to get the (double) Yangian. I'd imagine that, once you're there, you should be able to adapt the proof? But I don't have the references handy and haven't checked this myself, so this is just my intuition – Jules Lamers Jun 27 at 20:14
• @JulesLamers Thank you very much! I also feel this should work. I will get a try. – Zhihua Chang Jun 28 at 19:17