奇异自守积与BKM代数
Singular automorphic products and BKM algebras
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中文总结 AI 辅助
本文证明在共轭意义下，形如$2U\oplus L$的格上恰好有27个奇异权全纯自守积，通过BKM超代数与仿射李超代数分类，推广了Feingold-Frenkel的工作。
中文摘要 AI 辅助
在证明月光猜想之后，Borcherds于1995年提出：是否存在有限多个奇异权的全纯自守积？我们证明，在共轭意义下，在形如$2U\oplus L$的格上恰好有27个这样的积。分类过程通过将每个积在标准一维尖点处与某个BKM超代数的超分母等同起来进行，该BKM超代数带有仿射李超代数$\hat{\mathfrak{g}}$上的分次模结构。辅助代数$\hat{\mathfrak{g}}$被证明满足强条件，这些条件将可能性缩减到初始的1372个候选者。一个消去论证恰好留下85个与这27个积对应的代数$\hat{\mathfrak{g}}$。该分类与顶点算子超代数的某些特殊族密切相关。我们的结果为将仿射李超代数扩展为具有模性的双曲李超代数提供了系统框架，推广了Feingold和Frenkel（1983）的工作。
英文摘要
After proving the moonshine conjecture, Borcherds asked in 1995 whether there exist only finitely many holomorphic automorphic products of singular weight. We show that, up to conjugation, there are exactly 27 such products on lattices of the form $2U\oplus L$. The classification proceeds by identifying each product at a standard $1$-dimensional cusp with the superdenominator of a BKM superalgebra that carries a graded module structure over an affine Lie superalgebra $\hat{\mathfrak{g}}$. The auxiliary algebras $\hat{\mathfrak{g}}$ are shown to satisfy strong conditions that reduce the possibilities to an initial list of 1372 candidates. An elimination argument leaves exactly 85 algebras $\hat{\mathfrak{g}}$ that correspond to the 27 products. This classification is closely connected with certain distinguished families of vertex operator superalgebras. Our results provide a systematic framework for extending affine Lie superalgebras to hyperbolic ones with modularity, generalizing the work of Feingold and Frenkel (1983).
发表机构
- University of Science and Technology of China（中国科学技术大学）
- Wuhan University（武汉大学）
- Heidelberg University（海德堡大学）
机构由 AI 辅助整理，请以论文原文为准。
