Urod algebras and Translation of Walgebras
Abstract
In this work, we introduce Urod algebras associated to simplylaced Lie algebras as well as the concept of translation of Walgebras. 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 lefthandside the DrinfeldSokolov 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 Walgebras 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 NakajimaYoshioka blowup equations for the moduli space of framed torsion free sheaves on $\mathbb{CP}^2$ of an arbitrary rank.
 Publication:

arXiv eprints
 Pub Date:
 October 2020
 DOI:
 10.48550/arXiv.2010.02427
 arXiv:
 arXiv:2010.02427
 Bibcode:
 2020arXiv201002427A
 Keywords:

 Mathematics  Representation Theory;
 High Energy Physics  Theory;
 Mathematics  Quantum Algebra