Urod algebras and Translation of W-algebras

In this work, we introduce Urod algebras associated to simply-laced Lie algebras as well as the concept of translation of W-algebras. Both results are achieved by showing that the quantum Hamiltonian reduction commutes with tensoring with integrable representations, that is, for $V$ and $L$ an affine vertex algebra and an integrable affine vertex algebra associated with $\mathfrak{g}$, we have the vertex algebra isomorphism $H_{DS,f}^0(V\otimes L)\cong H_{DS,f}^0(V)\otimes L$, where in the left-hand-side the Drinfeld-Sokolov reduction is taken with respect to the diagonal action of $\widehat{\mathfrak{g}}$ on $V\otimes L$. The proof is based on some new constructionof automorphisms of vertex algebras, which may be of independent interest. As corollaries we get fusion categories of modules of many exceptional W-algebras and we can construct various corner vertex algebras. A major motivation for this work is that Urod algebras of type $A$ provide a representation theoretic interpretation of the celebrated Nakajima-Yoshioka blowup equations for the moduli space of framed torsion free sheaves on $\mathbb{CP}^2$ of an arbitrary rank.