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Article Dans Une Revue Advances in Mathematics Année : 2022

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

Résumé

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.

Dates et versions

hal-02467297 , version 1 (04-02-2020)

Identifiants

Citer

Rouven Frassek, Vasily Pestun, Alexander Tsymbaliuk. Lax matrices from antidominantly shifted Yangians and quantum affine algebras: A-type. Advances in Mathematics, 2022, 401, pp.108283. ⟨10.1016/j.aim.2022.108283⟩. ⟨hal-02467297⟩
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