Lattice vertex algebras over a field of positive characteristic: degenerate cases

Abstract

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 $V_{L,F}/\mathrm{Rad}$.

19 pages