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, we have the vertex algebra isomorphism, where in the left-hand-side the Drinfeld-Sokolov reduction is taken with respect to the diagonal action of on. The proof is based on some new construction of 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 of an arbitrary rank.