Uniform modular lattices and affine buildings

Abstract

In this paper, we present a simple lattice-theoretic characterization for affine buildings of type A. We introduce a class of modular lattices, called uniform modular lattices, and show that uniform modular lattices and affine buildings of type A constitute the same object. This is an affine counterpart of the well-known equivalence between projective geometries ($\simeq$ complemented modular lattices) and spherical buildings of type A.

arXiv commentver 2; according to the suggestion of the referee, Euclidean building was changed to affine building. To appear in Advances in Geometry