An Application of Collapsing Levels to the Representation Theory of Affine Vertex Algebras

• Published in: International mathematics research notices 2020-07, Vol.2020 (13), p.4103-4143
• Main Authors: Adamović, Dražen, Kac, Victor G, Möseneder Frajria, Pierluigi, Papi, Paolo, Perše, Ozren
• Format: Article
• Language: English
• ISSN: 1073-7928; 1687-0247
• DOI: 10.1093/imrn/rny237

Abstract We discover a large class of simple affine vertex algebras $V_{k} ({\mathfrak{g}})$, associated to basic Lie superalgebras ${\mathfrak{g}}$ at non-admissible collapsing levels $k$, having exactly one irreducible ${\mathfrak{g}}$-locally finite module in the category ${\mathcal O}$. In the case when ${\mathfrak{g}}$ is a Lie algebra, we prove a complete reducibility result for $V_k({\mathfrak{g}})$-modules at an arbitrary collapsing level. We also determine the generators of the maximal ideal in the universal affine vertex algebra $V^k ({\mathfrak{g}})$ at certain negative integer levels. Considering some conformal embeddings in the simple affine vertex algebras $V_{-1/2} (C_n)$ and $V_{-4}(E_7)$, we surprisingly obtain the realization of non-simple affine vertex algebras of types $B$ and $D$ having exactly one nontrivial ideal.