A vertex operator algebra $V_L$ associated with a positive definite even lattice $L$ has a standard integral form, which we denote it by $V_{L,\mathbb{Z}} $. If $F$ is a field of characteristic $p>0$, it is known that $V_{L,F}:= F\otimes_\mathbb{Z} V_{L,\mathbb{Z}}$, a vertex algebra over $F$, is simple if and only if $(p, \det(L))=1$. In this article, we study $V_{L,F}$ when the characteristic of $F$ divides $\det(L)$. We determine the radical $\mathrm{Rad}$ of the invariant bilinear form on $V_{L,F}$, show that it is the unique maximal ideal and study the quotient vertex algebra $V_{L,F}/\mathrm{Rad}$.
We study lattice vertex algebras of type ADE over an algebraically closed field $\mathbb{F}$ of prime characteristic $p>2$ and their representations. Let $L$ be a root lattice of type ADE, and $G_{L}$ the Gram matrix of $L$. When $\det G_{L}\not \equiv 0\pmod{p}$, we establish an isomorphism between the lattice vertex...
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...
We study regular $\mathbb{N}$-graded vertex operator algebras $V=\bigoplus_{n\geq 0}V_n$ whose weight-zero algebra $V_0$ is a nontrivial finite-dimensional local Gorenstein algebra, asking which structural features of strongly rational vertex operator algebras persist without the CFT-type condition $V_0=\mathbb{C}{\bf...
We prove an incidence bound for bipartite graphs on finite subsets of $\mathbb{F}^2\times \mathbb{F}^2$ defined by Boolean combinations of polynomial equations of bounded degree. If such a graph is $K_{k,k}$-free and its vertex classes have sizes $m$ and $n$, then it has $O_{t,k}((mn)^{2/3}+m+n+mn/p)$ edges, where $t$...
Let $G$ be a finite simple graph on $[n]$ and let $I_c(G)$ denote its complementary edge ideal in the polynomial ring $S = K[x_1,\dots,x_n]$. We give a combinatorial description, in terms of the structure of $G$, of the minimal generators of the symbolic Rees algebra $\mathcal{R}_s(I_c(G)) = \bigoplus_{k \geq 0} I_c(G)...
Antonino Ficarra, Somayeh Moradi, Y. Muta· 2 citations
We give an explicit example of a $W^{*}$-bundle $M$ over a compact, metrizable space $K$, which has each fiber $M_{p}$ a $\textrm{II}_{1}$-factor with separable predual, and which satisfies the following negation of the generator problem: given any finite family $a_{1},\cdots,a_{n}\in M$ of continuous sections, there i...
Ben Hayes· 0 citations
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