发表机构
Guangzhou Civil Aviation College（广州民航职业技术学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究正特征顶点算子代数的高阶Zhu诱导模的商结构，确定典范映射的核并给出同构条件，进而构造非分裂短正合序列。
AI 中文摘要
设$F$为特征不同于$2$的代数闭域，$V$为$F$上非负整数分次的顶点算子代数。我们考虑高阶Zhu代数，其通过取$V$关于包含向量$D_V^{(h)}a-\binom{-\mathrm{wt}(a)}{h}a$（对所有齐次$a\in V$及整数$h\geq1$成立，其中$D_V^{(h)}a=a_{-h-1}\mathbf1$）的关系空间的商得到。对于$n\geq1$及相应高阶Zhu代数上的非零幺模$U$，我们研究诱导模的商$L_n(U)$，并确定典范映射$U\to\Omega_n(L_n(U))/\Omega_{n-1}(L_n(U))$的核。该核是$U$中通过相邻低层代数分解的最大子模。当此子模为零时，典范映射为同构，且每个$\Omega_r$是前$r+1$个齐次子空间的直和。这些结果利用整数系数的零模展开及标准诱导模的分次证明。此外，该构造对模同态是自然的，且当$U$不可分解时，$L_n(U)$在分次模范畴中不可分解。从满足所述假设的高阶Zhu代数的模的非分裂扩张，我们构造弱$V$-模的非分裂短正合序列。
英文摘要
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 $D_V^{(h)}a-\binom{-\mathrm{wt}(a)}{h}a$ for all homogeneous $a\in V$ and integers $h\geq1$, where $D_V^{(h)}a=a_{-h-1}\mathbf1$. For $n\geq1$ 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 $U\toΩ_n(L_n(U))/Ω_{n-1}(L_n(U))$. 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.
