An application of collapsing levels to the representation theory of affine vertex algebras

TL;DR

This paper explores collapsing levels in affine vertex algebras linked to Lie superalgebras, revealing unique modules, reducibility, and ideal structures, with implications for conformal embeddings and algebra classifications.

Contribution

It introduces a broad class of simple affine vertex algebras at collapsing levels, characterizes module reducibility, and identifies generators of maximal ideals, advancing understanding of affine vertex algebra structures.

Findings

Existence of exactly one irreducible module in certain categories.

Complete reducibility of modules at collapsing levels for Lie algebras.

Identification of generators of maximal ideals at negative levels.

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 non-trivial ideal.