Urod algebras and Translation of W-algebras

Abstract

Abstract 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 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 $\mathbb {CP}^{2}$ of an arbitrary rank.