The Cayley Transform on Representations

TL;DR

This paper characterizes when the classical Cayley transform can be applied to irreducible representations of Lie groups, providing criteria, geometric conditions, and a classification for classical complex simple Lie groups and their compact real forms.

Contribution

It offers a representation theoretic generalization of the Cayley transform, establishing applicability criteria and classifying applicable representations for classical Lie groups.

Findings

Criteria for Cayley transform applicability to Lie group representations

Geometric conditions on weight diagrams for semisimple groups

Complete classification for classical complex simple Lie groups and their compact real forms

Abstract

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.