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We construct a homomorphism from the affine Yangian associated with sl (qᵤ-qₔ+₁) to the universal enveloping algebra of a W-algebra associated with gl (ₒ=₁ˡqₛ) and a nilpotent element of type (1^q₁-q₂, 2^q₂-q₃, , (l-1) ^q₋-₁-qₗ, l^qₗ) by using the coproduct and evaluation map for the affine Yangian and the Miura map for non-rectangular W-algebras. As an application, we give a homomorphism from the affine Yangian to some cosets of non-rectangular W-algebras.
Mamoru Ueda (Sun,) studied this question.