On the Structure of Higher Zhu Induced Modules in Positive Characteristic
Abstract
Let F be an algebraically closed field of characteristic different from $2$, and let V be a nonnegatively integer-graded vertex operator algebra over F. We consider higher Zhu algebras obtained by taking a quotient of V by a relation space containing the vectors DV(h)a−binom−wt(a)ha for all homogeneous a∈ V and integers h≥1, where DV(h)a=a−h−11. For n≥1 and a nonzero unital module U over the corresponding higher Zhu algebra, we study the quotients Ln(U) of induced modules and determine the kernel of the canonical map U→Ωn(Ln(U))/Ωn−1(Ln(U)). This kernel is the largest submodule of U that factors through the adjacent lower-level algebra. When this submodule is zero, the canonical map is an isomorphism, and each Ωr is the direct sum of the first r+1 homogeneous subspaces. These results are proved using an expansion of zero modes with integer coefficients and the grading of the standard induced modules. Moreover, the construction is natural with respect to module homomorphisms, and Ln(U) is indecomposable in the graded module category whenever U is indecomposable. From nonsplit extensions of modules for higher Zhu algebras satisfying the stated hypotheses, we construct nonsplit short exact sequences of weak V-modules.
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BibTeX
@misc{huang2026structure,
title = {{On the Structure of Higher Zhu Induced Modules in Positive Characteristic}},
author = {Haihua Huang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09523},
eprint = {2610.09523},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09523},
}
RIS
TY - UNPB
TI - On the Structure of Higher Zhu Induced Modules in Positive Characteristic
AU - Haihua Huang
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09523
UR - https://arxiv.org/abs/2610.09523
AB - Let F be an algebraically closed field of characteristic different from $2$, and let V be a nonnegatively integer-graded vertex operator algebra over F. We consider higher Zhu algebras obtained by taking a quotient of V by a relation space containing the vectors D_V^(h)a−binom−wt(a)ha for all homogeneous a∈ V and integers h≥1, where D_V^(h)a=a_−h−11. For n≥1 and a nonzero unital module U over the corresponding higher Zhu algebra, we study the quotients L_n(U) of induced modules and determine the kernel of the canonical map U→Ω_n(L_n(U))/Ω_n−1(L_n(U)). This kernel is the largest submodule of U that factors through the adjacent lower-level algebra. When this submodule is zero, the canonical map is an isomorphism, and each Ω_r is the direct sum of the first r+1 homogeneous subspaces. These results are proved using an expansion of zero modes with integer coefficients and the grading of the standard induced modules. Moreover, the construction is natural with respect to module homomorphisms, and L_n(U) is indecomposable in the graded module category whenever U is indecomposable. From nonsplit extensions of modules for higher Zhu algebras satisfying the stated hypotheses, we construct nonsplit short exact sequences of weak V-modules.
ER -