Mathematics > Representation Theory
[Submitted on 1 Jul 2022 (v1), last revised 27 Aug 2022 (this version, v2)]
Title:On rationality of $\mathbb{C}$-graded vertex algebras and applications to Weyl vertex algebras under conformal flow
View PDFAbstract:Using the Zhu algebra for a certain category of $\mathbb{C}$-graded vertex algebras $V$, we prove that if $V$ is finitely $\Omega$-generated and satisfies suitable grading conditions, then $V$ is rational, i.e. has semi-simple representation theory, with one dimensional level zero Zhu algebra. Here $\Omega$ denotes the vectors in $V$ that are annihilated by lowering the real part of the grading. We apply our result to the family of rank one Weyl vertex algebras with conformal element $\omega_\mu$ parameterized by $\mu \in \mathbb{C}$, and prove that for certain non-integer values of $\mu$, these vertex algebras, which are non-integer graded, are rational, with one dimensional level zero Zhu algebra. In addition, we generalize this result to appropriate $\mathbb{C}$-graded Weyl vertex algebras of arbitrary ranks.
Submission history
From: Florencia Orosz Hunziker [view email][v1] Fri, 1 Jul 2022 19:30:21 UTC (43 KB)
[v2] Sat, 27 Aug 2022 01:52:31 UTC (71 KB)
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