Feigin, B.Jimbo, M.Mukhin, E.Vilkovisky, I.2022-12-192022-12-192021-06Feigin, B., Jimbo, M., Mukhin, E., & Vilkoviskiy, I. (2021). Deformations of W algebras via quantum toroidal algebras. Selecta Mathematica, 27(4), 52. https://doi.org/10.1007/s00029-021-00663-01022-1824, 1420-9020https://hdl.handle.net/1805/30763We 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.en-USPublisher PolicyIntegrals of motion17B37Quantum toroidal algebraArticle