Acceptable compact Lie groups

TL;DR

This paper characterizes acceptable connected compact Lie groups, showing they must have derived subgroups isomorphic to specific classical groups, and explores invariant functions on certain complex groups.

Contribution

It provides a complete characterization of acceptable connected compact Lie groups based on their derived subgroups and investigates invariant functions on complex orthogonal groups.

Findings

Acceptable groups have derived subgroups isomorphic to specific classical groups.

Invariant functions on $ ext{SO}_4( ext{C})^2$ are not generated by 1-argument invariants.

$ ext{SO}_4( ext{C})$ is acceptable despite complex invariant function properties.

Abstract

In this paper we show that for a connected compact Lie group to be acceptable it is necessary and sufficient that its derived subgroup is isomorphic to a direct product of the groups $\SU(n)$, $\Sp(n)$, $\SO(2n+1)$, $\G_2$, $\SO(4)$. We show that there are invariant functions on $\SO_{4}(\mathbb{C})^{2}$ which are not generated by 1-argument invariants, though the group $\SO_{4}(\mathbb{C})$ is acceptable.