Summary:

The initial motivation behind Bruhat--Tits theory was to classify maximal bounded subgroups of reductive algebraic groups defined over a local field. In the same spirit, one goal of Project A5 is to get an analogous classification for maximal Lie algebra orders in the twisted loop algebra of a real simple Lie algebra. The interest in studying this problem originates from the theory of the classical Yang--Baxter equation. This field of research is closely related to another goal of this project, namely to develop foundations of Poisson geometry in the infinite-dimensional setting. Notably, trigonometric solutions of the classical Yang--Baxter equation enable one to define Lie--Poisson structures on loop groups. Affine buildings are also going to play a key role in the study of finiteness properties of the groups $\mathrm{SL}_n(Z[t,t^{-1}])$.