The Curtis–Tits-presentation

Abstract

Let B be an irreducible, spherical Moufang building of rank ⩾2, A an apartment of B and Φ the set of roots of A with corresponding root-subgroups A r⩽ Aut( B) , r∈ Φ. Let G=〈 A r | r∈ Φ〉 be the group of ‘Lie-type B ’. If Π is a fundamental system of Φ denote by X r =〈 A r , A − r 〉, r∈ Π and X rs =〈 X r , X s 〉 for r≠ s∈ Π. Then the following: Let Ĝ be the amalgamated product of the X rs , amalgamated over the X r and π : G ̂ →G the natural homomorphism. Then ker π⩽Z( G ̂ ) , is known as the Curtis–Tits-presentation for G. In this paper the proof of a generalization of this theorem, which uses only local information, is given.