AI 中文总结
本文研究素特征域$F$的特征整除格$L$的行列式时的格顶点代数$V_{L,F}$，确定其不变双线性型的根基为唯一极大理想，并研究对应的商顶点代数。
AI 中文摘要
与正定偶格$L$关联的顶点算子代数$V_L$具有标准整形式，记为$V_{L,\boldsymbol{Z}}$。若$F$是特征为$p>0$的域，已知$V_{L,F}:= F\bigotimes_\boldsymbol{Z} V_{L,\boldsymbol{Z}}$作为$F$上的顶点代数是单的当且仅当$(p, \text{det}(L))=1$。本文研究当$F$的特征整除$\text{det}(L)$时的$V_{L,F}$，确定了$V_{L,F}$上不变双线性型的根基$\text{Rad}$，证明其为唯一极大理想，并研究商顶点代数$V_{L,F}/\text{Rad}$。
英文摘要
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 $V_{L,F}/\mathrm{Rad}$.
Comments19 pages
