Curtis-Tits groups generalizing Kac-Moody groups of type A_n−1

In [13] we define a Curtis–Tits group as a certain generalization of a Kac–Moody group. We distinguish between orientable and nonorientable Curtis–Tits groups and identify all orientable Curtis–Tits groups as Kac–Moody groups associated to twin-buildings. In the present paper we construct all orientable as well as nonorientable Curtis–Tits groups with diagram An−1 (n 4) over a field k of size at least 4. The resulting groups are quite interesting in their own right. The orientable ones are related to Drinfeld’s construction of vector bundles over a non-commutative projective line and to the classical groups over cyclic algebras. The nonorientable
ones are related to expander graphs [14] and have symplectic, orthogonal and unitary groups as quotients.