On the Structure of Higher Zhu Induced Modules in Positive Characteristic

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 $D_V^{(h)}a-\left(\genfrac{}{}{0px}{}{-\mathrm{w}\mathrm{t}(a)}{h}\right)a$ for all homogeneous $a\in V$ and integers $h\ge 1$, where $D_V^{(h)}a=a_{-h-1}\mathbf{1}$. For $n\ge 1$ 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 $U\to {\mathrm{\Omega }}_n(L_n(U))\mathrm{/}{\mathrm{\Omega }}_{n-1}(L_n(U))$. 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 ${\mathrm{\Omega }}_r$ 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.