Curtis-Tits groups generalizing Kac-Moody groups of type $\widetilde{A}_n$

Details Curtis-Tits groups generalizing Kac-Moody groups of type $\widetilde{A}_n$ Curtis-Tits groups generalizing Kac-Moody groups of type $\widetilde{A}_n$

2009-11-04

arxiv.org/abs/0911.0824v2

Mathematics

Group Theory

Key Words: Curtis-Tits groups, Kac-Moody groups, twin-building, amalgam, opposite, Moufang foundations

Scientific paper

In a previous paper we define a Curtis-Tits group as a certain generalization of a Kac-Moody group. We distinguish between orientable and non-orientable 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 and non-orientable Curtis-Tits groups with diagram $\widetilde{A}_n$ over a field ${\mathbb F}$. The resulting groups are quite interesting in their own right. The orientable ones are related to Drinfel'd' s construction of vector bundles over a non-commutative projective line and to the classical groups over cyclic algebras. The non-orientable ones are related to q-CCR algebras in physics and have symplectic, orthogonal and unitary groups as quotients.

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