We study automorphic Lie algebras and their applications to integrable systems. Automorphic Lie algebras are a natural generalisation of celebrated Kac-Moody algebras to the case when the group of automorphisms is not cyclic. They are infinite dimensional and almost graded. We formulate the concept of a graded isomorphism and classify $sl(2,\mathbb{C})$ based automorphic Lie algebras corresponding to all finite reduction groups. We show that hierarchies of integrable systems, their Lax representations and master symmetries can be naturally formulated in terms of automorphic Lie algebras.

中文翻译：

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自守李代数和相应的可积系统

我们研究自立李代数及其在可积系统中的应用。自同构李代数是著名的Kac-Moody代数对自同构群不是循环的情况的自然概括。它们是无穷大的，几乎是渐变的。我们制定分级同构的概念并进行分类$s升（2，\mathbb{C}）$对应于所有有限约简组的基于自守的李代数。我们表明，可积系统的层次结构，它们的Lax表示和主对称性可以自然地用自构李式代数来表示。