Deformations of \mathcal {W} algebras via quantum toroidal algebras

• B. Feigin ^[3] ; M. Jimbo ^[1] ; E. Mukhin ^[2] ; I. Vilkoviskiy ^[4]
  1. [1] Rikkyo University

     Rikkyo University

     Japón

     
  2. [2] Purdue University

     Purdue University

     Township of Wabash, Estados Unidos

     
  3. [3] National Research University Higher School of Economics; Landau Institute for Theoretical Physics, Rusia
  4. [4] Skolkovo Institute of Science and Technology; National Research University Higher School of Economics, Rusia

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 27, Nº. 4, 2021
• Idioma: inglés
• DOI: 10.1007/s00029-021-00663-0
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• Resumen

  • We study the uniform description of deformed W algebras of type A including the supersymmetric case in terms of the quantum toroidal gl1 algebra E. In particular, we recover the deformed affine Cartan matrices and the deformed integrals of motion. We introduce a comodule algebra K over E which gives a uniform construction of basic deformed W currents and screening operators in types B,C,D including twisted and supersymmetric cases. We show that a completion of algebra K contains three commutative subalgebras. In particular, it allows us to obtain a commutative family of integrals of motion associated with affine Dynkin diagrams of all non-exceptional types except D(2)ℓ+1. We also obtain in a uniform way deformed finite and affine Cartan matrices in all classical types together with a number of new examples, and discuss the corresponding screening operators.