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

Authors: Dražen Adamović, Victor G Kac, Pierluigi Möseneder Frajria, Paolo Papi, Ozren Perše

International Mathematics Research Notices, 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 $V_{-\frac{1}{2}}(C_n)$ 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.

Pages