发表机构
Yale University（耶鲁大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过将BKM代数中的无限维Heisenberg代数替换为Virasoro代数，构造了VBKM代数族，并证明其具有类似的分母恒等式和特征公式，且将假怪兽李代数纳入该族。
AI 中文摘要
Borcherds-Kac-Moody (BKM) 李代数是通过以特定方式组合若干 $\mathfrak{sl}_2$ 和三维Heisenberg代数而构造的，推广了著名的Kac-Moody代数。这类代数有广泛的应用，包括怪兽和假怪兽李代数的例子都是BKM代数。它们的主要特征包括不变双线性型的存在性、分母恒等式以及不可约表示的Weyl-Kac特征公式。我们引入了一种密切相关代数的构造，称之为Virasoro-Borcherds-Kac-Moody (VBKM) 代数：如果BKM代数内部的某些三维Heisenberg代数恰好形成无限维Heisenberg代数，我们在构造中将其替换为Virasoro代数。我们在 $\mathbb{P}^1$ 上构造了一族李代数，其中一般纤维是VBKM代数，特殊纤维是BKM代数。这使我们能够将VBKM代数视为BKM代数的形变，并证明VBKM代数具有类似于BKM代数的结构性质，尽管缺乏不变双线性型。这些性质包括分母恒等式和不可约表示的特征公式。我们给出了在格顶点算子代数中实现这种 $\mathbb{P}^1$ 族VBKM代数的例子。特别地，我们将假怪兽李代数包含在这样的族中。
英文摘要
Borcherds-Kac-Moody (BKM) Lie algebras are constructed by combining a number of $\mathfrak{sl}_2$-s and $3$-dimensional Heisenberg algebras in a certain way, generalizing the celebrated Kac-Moody algebras. Such algebras have a wide range of applications, including the examples of the monster and fake monster Lie algebras being BKM algebras. Among their main features are existence of an invariant bilinear form, denominator identity and Weyl-Kac character formulas for irreducible representations. We introduce a construction of closely related algebras that we call Virasoro-Borcherds-Kac-Moody (VBKM) algebras: if it happens that certain $3$-dimensional Heisenberg algebras inside a BKM algebra form the infinite-dimensional Heisenberg algebra, we replace it by the Virasoro algebra in the construction. We construct a family of Lie algebras over $\mathbb{P}^1$, where the generic fiber is a VBKM algebra and a special fiber is a BKM algebra. This allows us to visualize VBKM algebras as deformations of BKM algebras and prove structural properties about VBKM algebras similar to those of BKM algebras, despite the absence of an invariant bilinear form. Those include denominator identity and character formulas of irreducible representations. We give examples of realizations of such $\mathbb{P}^1$-families of VBKM algebras in lattice vertex operator algebras. In particular, we include the fake monster Lie algebra into such a family.
Comments29 pages. Changes: A couple of missing conditions added to Definition 3.1 and the last condition got relaxed. Arguments in Remark 3.7 simplified. This made Proposition 3.5 redundant and removed. Remark 3.7 changed into Computation 3.6. A non-trivial ideal found in non-generic VBKM algebras and given in Proposition 3.6 and illustrated in Example 5.8. Conjecture 4.8 got adapted to this new discovery
