Two Homomorphisms from the affine Yangian associated with sl(n) to the affine Yangian associated with sl(n+1)

We construct a homomorphism from the affine Yangian Y,+(sl(n)) to the affine Yangian Y,(sl(n+1)) which is different from the one in arXiv:2312.09933. By using this homomorphism, we give a homomorphism from Y,(sl(n))⊗ Y,+n(sl(m)) to Y,(sl(m+n)). As an application, we construct a homomorphism from the affine Yangian Y,+n(sl(m)) to the centralizer algebra of the pair of affine Lie algebras (gl(m+n),gl(n)) and the coset vertex algebra of the pair of rectangular W-algebras (Wᵏ(gl(2m+2n),(2ᵐ⁺ⁿ)),Wᵏ⁺ᵐ(gl(2n),(2ⁿ))).