表示上的凯莱变换
The Cayley Transform on Representations
AI总结：
本文从表示论角度推广经典凯莱变换，建立一般李群及半单李群上适用性的判据和权图几何条件，并完成经典复单李群及其紧实形式的完整分类，发现除已知例子外仅有Spin(8)的自旋表示适用。
摘要：
经典凯莱变换是二次矩阵群与其李代数之间的双有理映射，由凯莱于1846年首次发现。由于其在纯数学和应用数学中的重要作用，经典凯莱变换已从多种视角得到推广。本文关注经典凯莱变换的表示论推广。本工作背后的思想是：经典凯莱变换的适用性在很大程度上取决于李群如何被表示。目标是刻画李群的不可约表示，使得经典凯莱变换对其适用。为此，我们首先建立一般李群适用性的判据。若该群是半单的，我们进一步得到此类表示的权图上的几何条件。最后，我们对经典复单李群及其紧实形式给出完整分类。除先前已知的例子外，$\mathrm{Spin}(8)$ 的自旋表示是清单中仅有的新例子。
英文摘要：
The classical Cayley transform is a birational map between a quadratic matrix group and its Lie algebra, which was first discovered by Cayley in 1846. Because of its essential role in both pure and applied mathematics, the classical Cayley transform has been generalized from various perspectives. This paper is concerned with a representation theoretic generalization of the classical Cayley transform. The idea underlying this work is that the applicability of the classical Cayley transform heavily depends on how the Lie group is represented. The goal is to characterize irreducible representations of a Lie group, to which the classical Cayley transform is applicable. To this end, we first establish criteria of the applicability for a general Lie group. If the group is semisimple, we further obtain a geometric condition on the weight diagram of such representations. Lastly, we provide a complete classification for classical complex simple Lie groups and their compact real forms. Except for the previously known examples, spin representations of $\mathrm{Spin}(8)$ are the only ones on our list.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
