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 $$\widehat{\mathfrak {sl}}(n)$$ 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 [6]. 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. It is expected that this homomorphism will be helpful for the resolution of the Crutzig-Diaconescu-Ma’s conjecture.