Commuting Subalgebras of Affine Super Yangians Arising from Edge Contractions

In the previous paper, we constructed two kinds of edge contractions for the affine super Yangian and a homomorphism from the affine super Yangian to the universal enveloping algebra of a $W$-superalgebra of type $A$. In this article, we show that these two edge contractions commute with each other. As an application, we give a homomorphism from the affine super Yangian to some centralizer algebras of the universal enveloping algebra of $W$-superalgebras of type $A$. Using the edge contraction, we also show the compatibility of the coproduct for the affine super Yangian with the parabolic induction for a $W$-superalgebra of type $A$ in some special cases.

💡 Research Summary

The paper studies two edge‑contraction homomorphisms that were introduced in the author’s earlier work for the affine super Yangian associated with the Lie superalgebra $\mathfrak{bsl}(m|n)$. The first contraction $\Psi_{1}$ embeds the Yangian $Y_{\hbar,\varepsilon}(\mathfrak{bsl}(m_{1}|n_{1}))$ into the standard degreewise completion $eY_{\hbar,\varepsilon}(\mathfrak{bsl}(m_{1}+m_{2}|n_{1}+n_{2}))$, while the second contraction $\Psi_{2}$ does the same for a shifted Yangian $Y_{\hbar,\varepsilon+(m_{1}-n_{1})\hbar}(\mathfrak{bsl}(m_{2}|n_{2}))$. Both maps are given explicitly on the generators $X^{\pm}{i,r}$ and $H{i,r}$ by linear combinations of matrix units $E_{i,j}$ together with correction terms $P_{i},Q_{i}$ that involve infinite sums over the auxiliary indices.

The main result, Theorem 1.1, proves that the images of $\Psi_{1}$ and $\Psi_{2}$ commute inside the completed algebra. The proof does not rely on the current presentation of the affine super Yangian (as is common for quantum toroidal algebras) but instead uses the finite presentation introduced in earlier work. By checking the defining relations for all generators, the author shows $