Two homomorphisms from the affine Yangian associated with \(\widehat{\mathfrak {sl}}(n)\) to the affine Yangian associated with \(\widehat{\mathfrak {sl}}(n+1)\)

We construct a homomorphism from the affine Yangian \(Y_{\hbar ,\varepsilon +\hbar }(\widehat{\mathfrak {sl}}(n))\) $Y_{ħ,\epsilon +ħ}(\widehat{\mathfrak{sl}}(n))$ to the affine Yangian \(Y_{\hbar ,\varepsilon }(\widehat{\mathfrak {sl}}(n+1))\) $Y_{ħ,\epsilon }(\widehat{\mathfrak{sl}}(n+1))$ which is different from the one in Ueda (A homomorphism from the affine Yangian \(Y_{\hbar ,\varepsilon }(\widehat{\mathfrak {sl}}(n))\) $Y_{ħ,\epsilon }(\widehat{\mathfrak{sl}}(n))$ to the affine Yangian \(Y_{\hbar ,\varepsilon }(\widehat{\mathfrak {sl}}(n+1))\) $Y_{ħ,\epsilon }(\widehat{\mathfrak{sl}}(n+1))$ , 2023. arXiv:2312.09933). By using this homomorphism, we give a homomorphism from \(Y_{\hbar ,\varepsilon }(\widehat{\mathfrak {sl}}(n))\otimes Y_{\hbar ,\varepsilon +n\hbar }(\widehat{\mathfrak {sl}}(m))\) $Y_{ħ,\epsilon }(\widehat{\mathfrak{sl}}(n))\otimes Y_{ħ,\epsilon +nħ}(\widehat{\mathfrak{sl}}(m))$ to \(Y_{\hbar ,\varepsilon }(\widehat{\mathfrak {sl}}(m+n))\) $Y_{ħ,\epsilon }(\widehat{\mathfrak{sl}}(m+n))$ . As an application, we construct a homomorphism from the affine Yangian \(Y_{\hbar ,\varepsilon +n\hbar }(\widehat{\mathfrak {sl}}(m))\) $Y_{ħ,\epsilon +nħ}(\widehat{\mathfrak{sl}}(m))$ to the centralizer algebra of the pair of affine Lie algebras \((\widehat{\mathfrak {gl}}(m+n),\widehat{\mathfrak {sl}}(n))\) $(\widehat{\mathfrak{gl}}(m+n),\widehat{\mathfrak{sl}}(n))$ and the coset vertex algebra of the pair of rectangular W-algebras \(\mathcal {W}^k(\mathfrak {gl}(2m+2n),(2^{m+n}))\) ${\mathcal{W}}^k(\mathfrak{gl}(2m+2n),(2^{m+n}))$ and \(\mathcal {W}^{k+m}(\mathfrak {sl}(2n),(2^{n}))\) ${\mathcal{W}}^{k+m}(\mathfrak{sl}(2n),(2^n))$ .