2020/01/31 by Rouven Frassek, Vasily Pestun, Alexander Tsymbaliuk · 1 citation
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Affine transformation #Algebra over a field #Algebraic structures and combinatorial models #Mathematical analysis #Mathematics #Nonlinear Waves and Solitons #Physics #Pure mathematics #Quantum #Quantum mechanics #Realization (probability) #Simple (philosophy) #Trigonometry #Type (biology) #Yangian #math-ph #math.MP #math.RT
paper · pdf · doi:10.1016/j.aim.2022.108283
published as Advances in Mathematics 401 (2022), Paper No. 108283, 73pp · v2: 57 pages, typos fixed, some details and references added. v1: 54 pages, comments are welcome!
openalex publication_date 2020/02/04 · arxiv created 2022/02/22 · arxiv updated 2022/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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(\mathfrakgln) and quantum affine algebras Uμ+,μ-(L\mathfrakgln), which slightly generalize their \mathfraksln-counterparts. Our key observation is that both algebras admit the RTT type realization when μ (respectively, 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 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 \mathfraksln, previously established 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.