Defining amalgams of compact Lie groups

TL;DR

This paper characterizes groups generated by certain rank-one and rank-two subgroups related to compact Lie groups, showing they are central quotients of the corresponding simply connected compact Lie groups.

Contribution

It establishes that groups with a weak Phan system of a given Dynkin diagram type are central quotients of the associated simply connected compact Lie group.

Findings

Groups with weak Phan systems are central quotients of corresponding Lie groups.

The structure of these groups is determined by the Dynkin diagram type.

The result links algebraic group structures with Lie group classifications.

Abstract

For $n \geq 2$ let $\Delta$ be a Dynkin diagram of rank $n$ and let $I = {1, >..., n}$ be the set of labels of $\Delta$. A group $G$ admits a weak Phan system of type $\Delta$ over $\mathbb{C}$ if $G$ is generated by subgroups $U_i$, $i \in I$, which are central quotients of simply connected compact semisimple Lie groups of rank one, and contains subgroups $U_{i,j} = \gen{U_i ,U_j}$, $i \neq j \in I$, which are central quotients of simply connected compact semisimple Lie groups of rank two such that $U_i$ and $U_j$ are rank one subgroups of $U_{i,j}$ corresponding to a choice of a maximal torus and a fundamental system of roots for $U_{i,j}$. It is shown in this article that $G$ then is a central quotient of the simply connected compact semisimple Lie group whose complexification is the simply connected complex semisimple Lie group of type $\Delta$.