The coproduct for the affine Yangian and the parabolic induction for non-rectangular W-algebras

By using the coproduct and evaluation map for the affine Yangian and the Miura map for non-rectangular W-algebras, we construct a homomorphism from the affine Yangian associated with sl (n) to the universal enveloping algebra of a non-rectangular W-algebra of type A, which is an affine analogue of the one given in De Sole-Kac-Valeri. As a consequence, we find that the coproduct for the affine Yangian is compatible with the parabolic induction for non-rectangular W-algebras via this homomorphism. We also show that the image of this homomorphism is contained in the affine coset of the W-algebra in the special case that the W-algebra is associated with a nilpotent element of type (1^m-n, 2ⁿ).