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.

我们研究了变形的\(\mathcal {W}\)类型\(\textsf {A }\)代数的统一描述，包括超对称情况下的量子环面\({\mathfrak {g}}{\mathfrak {l}}_1\)代数\({{\mathcal {E}}}\)。特别是，我们恢复了变形的仿射 Cartan 矩阵和变形的运动积分。我们在\({{\mathcal {E}}}\)上引入了一个余模代数\(\mathcal {K}\)，它给出了基本变形\(\mathcal {W}\)电流和屏蔽算子的统一构造类型\(\textsf {B},\textsf {C},\textsf {D}\)包括扭曲和超对称情况。我们证明代数的完成\(\mathcal {K}\)包含三个交换子代数。特别是，它允许我们获得与除\(\textsf {D}^{(2)}_{\ell +1}\)之外的所有非异常类型的仿射 Dynkin 图相关的运动积分的交换族。我们还以统一的方式获得了所有经典类型的变形有限和仿射 Cartan 矩阵以及一些新示例，并讨论了相应的筛选算子。