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An application of collapsing levels to the representation theory of affine vertex algebras. (English) Zbl 07236279
Summary: 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.
##### MSC:
 17 Nonassociative rings and algebras 16 Associative rings and algebras
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