Rank one groups and division pairs

Abstract

Rank one groups were introduced by F. G. Timmesfeld as the building blocks of Lie type groups. Division pairs are algebraic objects categorically equivalent to Moufang sets. We define a functor $\varDelta$ from rank one groups to division pairs and prove that $\varDelta$ has a left adjoint $\varSigma$, given by a Steinberg type construction. We also extend the theory of the quasi-inverse and the Bergmann operators, well known from Jordan pairs, to this setting. As an application, we show that identities proved by T. De Medts and Y. Segev for Moufang sets hold in arbitrary rank one groups.