Affine Yangians and some cosets of non-rectangular W-algebras

We construct a homomorphism from the affine Yangian associated with sl(qᵤ-qᵤ₊₁) to the universal enveloping algebra of a W-algebra associated with gl(∑ₛ₌₁ˡqₛ) and a nilpotent element of type (1q₁-q₂,2q₂-q₃,,(l-1)^qₗ₋₁-qₗ,lqₗ) 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.