We show that automorphic Lie algebras which contain a Cartan subalgebra with a constant-spectrum, called hereditary, are completely described by 2-cocycles on a classical root system taking only two different values. This observation suggests a novel approach to their classification. By determining the values of the cocycles on opposite roots, we obtain the Killing form and the abelianization of the automorphic Lie algebra. The results are obtained by studying equivariant vectors on the projective line. As a byproduct, we describe a method to reduce the computation of the infinite-dimensional space of said equivariant vectors to a finite-dimensional linear computation and the determination of the ring of automorphic functions on the projective line.

中文翻译：

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遗传自守李代数

我们证明了包含具有恒定谱的 Cartan 子代数的自守李代数，称为遗传，完全由经典根系统上的 2-cocycles 描述，仅取两个不同的值。这一观察结果表明了一种新的分类方法。通过确定对根上的余环的值，我们得到了自守李代数的 Killing 形式和阿贝尔化。结果是通过研究投影线上的等变向量获得的。作为副产品，我们描述了一种将所述等变向量的无限维空间的计算简化为有限维线性计算和确定投影线上的自守函数环的方法。