We construct a homomorphism from the affine Yangian associated with sl̂(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 W-algebras. As an application, we give a homomorphism from the affine Yangian to some cosets of non-rectangular W-algebras.
Mamoru Ueda (Mon,) studied this question.