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Morita equivalences for Zhu's algebra

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arxiv 2403.11855 v3 pith:OGYNAUYE submitted 2024-03-18 math.RT math.QA

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abstract

Through the introduction of new ideals, and with the assistance of the $d$-th mode transition algebras $\mathfrak{A}_d$, for $d\in \mathbb{N}$, we show how Zhu's associative algebra $\mathsf{A}$, conventionally valued for tracking information about the degree $0$ part of an $\mathbb{N}$-graded module over a vertex operator algebra $V$, also contains information about components of higher degree. As an application, equivalent conditions are given for rationality of $V$, and explicit presentations for higher-level Zhu algebras are given, including for a large class of non-rational VOAs.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Modular functors from conformal blocks of rational vertex operator algebras

    math.QA 2025-07 conditional novelty 7.0 of 10

    Spaces of conformal blocks of a strongly rational vertex operator algebra form a modular functor, giving the module category a modular fusion structure and a 3D topological field theory extension.

  2. On the Strong Unital Property for the Affine VOAs

    math.QA 2026-01 reject novelty 6.0 of 10

    Universal affine sl₂ vacuum VOAs are never strongly unital at k ≠ −2; for L_{ŝl₂}(1,0) an explicit strong-unit formula is given, but it fails for d ≥ 2 under the paper's own affine bracket conventions.

  3. How are pseudo-$q$-traces related to (co)ends?

    math.QA 2025-08 conditional novelty 6.0 of 10

    The pseudo-q-trace construction is shown to be the same, through the sewing-factorization theorem, as the categorical end over the module category, proving conjectures of Gainutdinov-Runkel and Arike-Nagatomo.

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