Drinfeld proved that the Yangian Yg of a complex semisimple Lie
algebra g gives rise to solutions of the quantum Yang-Baxter equations
on irreducible, finite-dimensional representations of Yg, which are
rational in the spectral parameter. This result was recently extended
by Maulik-Okounkov for Kac-Moody algebras corresponding to
quivers, and representations arising from geometry.
Surprisingly perhaps, this rationality ceases to hold if one considers
arbitrary finite-dimensional representations of Yg, at least if one
requires such solutions to be natural with respect to the representation
and compatible with tensor products.
I will explain how one can instead produce meromorphic solutions
of the QYBE on all finite-dimensional representations by resumming
Drinfeld’s universal R-matrix R(s) of Yg. The construction hinges on
resumming the abelian part of R(s), and on realizing its lower triangular
part as a twist conjugating the standard coproduct of Yg to its deformed
Drinfeld coproduct.
This is joint work with Sachin Gautam and Curtis Wendlandt, and
is based on arXiv:1907.03525