We investigate a class of Kac-Moody algebras previously not considered. We refer to them as n-extended Lorentzian Kac-Moody algebras defined by their Dynkin diagrams through the connection of an $A_n$ Dynkin diagram to the node corresponding to the affine root. The cases $n=1$ and $n=2$ correspond to the well studied over and very extended Kac-Moody algebras, respectively, of which the particular examples of $E_{10}$ and $E_{11}$ play a prominent role in string and M-theory. We construct closed generic expressions for their associated roots, fundamental weights and Weyl vectors. We use these quantities to calculate specific constants from which the nodes can be determined that when deleted decompose the n-extended Lorentzian Kac-Moody algebras into simple Lie algebras and Lorentzian Kac-Moody algebra. The signature of these constants also serves to establish whether the algebras possess $SO(1,2)$ and/or $SO(3)$-principal subalgebras.

中文翻译：

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n-扩展洛伦兹 Kac-Moody 代数

我们研究了以前没有考虑过的一类 Kac-Moody 代数。我们将它们称为 n 扩展 Lorentzian Kac-Moody 代数，由它们的 Dynkin 图定义，通过 $A_n$ Dynkin 图连接到对应于仿射根的节点。案例 $n=1$ 和 $n=2$ 分别对应于经过充分研究和非常扩展的 Kac-Moody 代数，其中 $E_{10}$ 和 $E_{11}$ 的特定示例起到了在弦理论和 M 理论中的突出作用。我们为其关联的根、基本权重和 Weyl 向量构建封闭的通用表达式。我们使用这些量来计算特定常数，从中可以确定节点，当删除时，将 n 扩展的 Lorentzian Kac-Moody 代数分解为简单的 Lie 代数和 Lorentzian Kac-Moody 代数。