Defining Amalgams of Compact Lie Groups

R. Gramlich

Journal of Lie theory, Tome 16 (2006) no. 1, pp. 001-018

Résumé

For $n \geq 2$ let $\Delta$ be a Dynkin diagram of rank $n$ and let $I = \{ 1, \ldots, n \}$ be the set of labels of $\Delta$. A group $G$ admits a {\it weak Phan system of type $\Delta$ over $\C$} 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 $U_{i,j} = \langle U_i,U_j\rangle$, $i \neq j \in I$, 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 $\Delta$.

Détail

Classification : 22C05, 51E24, 20E42
Mots-clés : Compact Lie groups, Tits buildings, Phan-type theorems, amalgam method