发表机构
Illinois State University（伊利诺伊州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究缺少CFT型条件时正则N-分次顶点算子代数中仿射与格结构的保持，证明在若干假设下存在共形嵌入的格顶点算子代数，并指出正则性不足以保证半单性。
AI 中文摘要
我们研究正则 $\mathbb{N}$-分次顶点算子代数 $V=\bigoplus_{n\geq 0}V_n$，其零权重代数 $V_0$ 是非平凡有限维局部 Gorenstein 代数，探讨在缺少 CFT 型条件 $V_0=\mathbb{C}{\bf 1}$ 时，强有理顶点算子代数的哪些结构特征仍然保持。对于左 Leibniz 代数 $V_1$ 的半单李子代数，在假设 $\ker\left(\left.L(-1)\right|_{V_0}\right)=\mathbb{C}\mathbf{1}$ 下，乘积 $u_1v$ 诱导一个不变对称双线性型。在 $C_2$-余有限性和非退化条件下，每个具有非零型的简单分量生成正整数水平的仿射顶点算子代数，并在 $V$ 上可积地作用。当 $V_1$ 可解时，$V_0$ 的 Frobenius 结构产生一个特定的非退化子空间 $M\subset V_1$。在拟主条件（quasi-primary condition）下，$M$ 是阿贝尔的并生成一个海森堡顶点算子代数。在额外的半单性、满秩整性和余循环相容性假设下，$V$ 包含一个共形嵌入的格顶点算子代数 $V_K$，其中 $K$ 是正定偶格，秩为 $\dim_{\mathbb{C}}M$，最小范数至少为 $4$。共形平移的格顶点算子代数提供了显式模型，用以说明权重一李结构、模生成李结构与格结构之间的区别。它们还表明，仅正则性并不能保证任意海森堡零模作用的半单性，因此格定理的额外假设代表了真正的结构障碍。
英文摘要
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 $V_0=\mathbb{C}{\bf 1}$. For semisimple Lie subalgebras of the left Leibniz algebra $V_1$, assuming $\ker\left(\left.L(-1)\right|_{V_0}\right)=\mathbb{C}\mathbf{1}$, the product $u_1v$ induces an invariant symmetric bilinear form. Under $C_2$-cofiniteness and a nondegeneracy condition, each simple component with nonzero form generates an affine vertex operator algebra at positive integral level and acts integrably on $V$. When $V_1$ is solvable, the Frobenius structure of $V_0$ yields a distinguished nondegenerate subspace $M\subset V_1$. Under a quasi-primary condition, $M$ is abelian and generates a Heisenberg vertex operator algebra. With additional semisimplicity, full-rank integrality, and cocycle compatibility assumptions, $V$ contains a conformally embedded lattice vertex operator algebra $V_K$, where $K$ is positive-definite and even, with rank $\dim_{\mathbb{C}}M$ and minimum norm at least $4$. Conformally shifted lattice vertex operator algebras provide explicit models illustrating the distinction among weight-one Lie, mode-generated Lie, and lattice structures. They also show that regularity alone does not ensure semisimplicity of arbitrary Heisenberg zero-mode actions, so the additional lattice-theorem hypotheses represent genuine structural obstructions.
