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

TL;DR

This paper constructs and analyzes all orientable and non-orientable Curtis-Tits groups of type tilde{A}_n over a field, revealing connections to Kac-Moody groups, vector bundles, cyclic algebras, and q-CCR algebras.

Contribution

It provides a comprehensive construction of Curtis-Tits groups of type tilde{A}_n, extending previous work and establishing links to various mathematical and physical structures.

Findings

All orientable Curtis-Tits groups are identified as Kac-Moody groups associated to twin-buildings.

Non-orientable groups relate to q-CCR algebras and have classical groups as quotients.

The constructed groups connect to vector bundles over non-commutative projective lines and cyclic algebras.

Abstract

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.