Knibbeler Vincent, Lombardo Sara, Sanders Jan A
1Department of Mathematics, Faculty of Sciences, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands.
2Department of Mathematics and Information Sciences, Northumbria University, Newcastle upon Tyne, NE1 8ST UK.
Found Comut Math. 2017;17(4):987-1035. doi: 10.1007/s10208-016-9312-1. Epub 2016 Apr 11.
The paper presents the complete classification of Automorphic Lie Algebras based on , where the symmetry group is finite and acts on by inner automorphisms, has no trivial summands, and where the poles are in any of the exceptional -orbits in . A key feature of the classification is the study of the algebras in the context of . This provides on the one hand a powerful tool from the computational point of view; on the other, it opens new questions from an algebraic perspective (e.g. structure theory), which suggest further applications of these algebras, beyond the context of integrable systems. In particular, the research shows that this class of Automorphic Lie Algebras associated with the groups (tetrahedral, octahedral and icosahedral groups) depend on the group through the automorphic functions only; thus, they are group independent as Lie algebras. This can be established by defining a for these algebras, generalising this classical notion to the case of Lie algebras over a polynomial ring.
本文给出了基于 的自守李代数的完整分类，其中对称群 是有限的，并且通过内自同构作用于 ， 没有平凡直和项，且极点位于 中的任何例外 -轨道。该分类的一个关键特征是在 的背景下对这些代数进行研究。这一方面从计算的角度提供了一个强大的工具；另一方面，它从代数角度（例如结构理论）提出了新问题，这暗示了这些代数在可积系统背景之外的进一步应用。特别地，研究表明与 群（四面体群、八面体群和二十面体群）相关联的这类自守李代数仅通过自守函数依赖于群；因此，作为李代数它们与群无关。这可以通过为这些代数定义一个 来建立，将这个经典概念推广到多项式环上的李代数情形。
