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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.
17 Nonassociative rings and algebras
16 Associative rings and algebras
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