Lax matrices from antidominantly shifted Yangians and quantum affine algebras: A-type

Abstract

We construct a family of GLn rational and trigonometric Lax matrices TD(z) parametrized by Λ+-valued divisors D on P1. To this end, we study the shifted Drinfeld Yangians Yμ(gln) and quantum affine algebras Uμ+,μ−(Lgln), which slightly generalize their sln-counterparts of [3,18]. Our key observation is that both algebras admit the RTT type realization when μ (resp. μ+ and μ−) are antidominant coweights. We prove that TD(z) are polynomial in z (up to a rational factor) and obtain explicit simple formulas for those linear in z. This generalizes the recent construction by the first two authors of linear rational Lax matrices [15] in both trigonometric and higher z-degree directions. Furthermore, we show that all TD(z) are normalized limits of those parametrized by D supported away from {∞} (in the rational case) or {0,∞} (in the trigonometric case). The RTT approach provides conceptual and elementary proofs for the construction of the coproduct homomorphisms on shifted Yangians and quantum affine algebras of sln, previously established in [14,18] via rather tedious computations. Finally, we establish a close relation between a certain collection of explicit linear Lax matrices and the well-known parabolic Gelfand-Tsetlin formulas.