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, +₍ (sl (m) ) to Y, (sl (m+n) ). As an application, we construct a homomorphism from the affine Yangian Y, +₍ (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), (2^m+n) ), W^k+m (gl (2n), (2^n) ) ).