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

Abstract

We study the uniform description of deformed \(\mathcal {W}\) algebras of type \(\textsf {A}\) including the supersymmetric case in terms of the quantum toroidal \({\mathfrak {g}}{\mathfrak {l}}_1\) algebra \({{\mathcal {E}}}\). In particular, we recover the deformed affine Cartan matrices and the deformed integrals of motion. We introduce a comodule algebra \(\mathcal {K}\) over \({{\mathcal {E}}}\) which gives a uniform construction of basic deformed \(\mathcal {W}\) currents and screening operators in types \(\textsf {B},\textsf {C},\textsf {D}\) including twisted and supersymmetric cases. We show that a completion of algebra \(\mathcal {K}\) 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 \(\textsf {D}^{(2)}_{\ell +1}\). 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.