Shuffle Products, Degenerate Affine Hecke Algebras, and Quantum Toda Lattice

2025/05/14 by Kalmykov, Artem
#FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2505.09520

Abstract

We revisit an identification of the quantum Toda lattice for GL_N and the truncated shifted Yangian of \mathfraksl₂, as well as related constructions, from a purely algebraic point of view, bypassing the topological medium of the homology of the affine Grassmannian. For instance, we interpret the Gerasimov-Kharchev-Lebedev-Oblezin homomorphism into the algebra of difference operators via a finite analog of the Miura transform. This algebraic identification is deduced by relating degenerate affine Hecke algebras to the simplest example of a rational Feigin-Odesskii shuffle product. As a bonus, we obtain a presentation of the latter via a mirabolic version of the Kostant-Whittaker reduction.

Citations

• A categorical approach to dynamical quantum groups
• Differential operators on G/U and the Gelfand-Graev action
• Comultiplication for shifted Yangians and quantum open Toda lattice
• An introduction to affine Grassmannians and the geometric Satake equivalence
• Quantization of Drinfeld Zastava in type A
• Equivariant Satake category and Kostant-Whittaker reduction
• Holomorphic Principal Bundles Over Elliptic Curves II: The Parabolic Construction
• PBW and duality theorems for quantum groups and quantum current algebras
• Yangians and transvector algebras
• Residue construction of Hecke algebras
• Vector bundles on elliptic curve and Sklyanin algebras

Related

• Shuffle algebras, lattice paths and quantum toroidal \mathfrakgl_n|m
• Comultiplication for shifted Yangians and quantum open Toda lattice
• Quantum toroidal and shuffle algebras
• A Fourier transform for the quantum Toda lattice
• Quantum Toda Lattice: a Challenge for Representation Theory