Curtis-Tits Groups of simply-laced type

TL;DR

This paper classifies and constructs universal completions of Curtis-Tits groups of simply-laced type, extending known results to non-orientable cases and connecting them to twisted Kac-Moody groups.

Contribution

It generalizes the classification and realization of Curtis-Tits groups of simply-laced type, including non-orientable cases, as central extensions of twisted Kac-Moody groups.

Findings

Universal completions are central extensions of Kac-Moody groups.

Non-orientable amalgams relate to symmetry groups of unitary forms.

Extended classification covers all simply-laced diagrams.

Abstract

The classification of Curtis-Tits amalgams with {connected}, triangle free, simply-laced diagram over a field of size at least $4$ was completed in~\cite{BloHof2014b}. Orientable amalgams are those arising from applying the Curtis-Tits theorem to groups of Kac-Moody type, and indeed, their universal completions are central extensions of those groups of Kac-Moody type. The paper~\cite{BloHof2014a} exhibits concrete (matrix) groups as completions for all Curtis-Tits amalgams with diagram $\widetilde{A}_{n-1}$. For non-orientable amalgams these groups are symmetry groups of certain unitary forms over a ring of skew Laurent polynomials. In the present paper we generalize this to all amalgams arising from the classification above and, under some additional conditions, exhibit their universal completions as central extensions of twisted groups of Kac-Moody type.