Uniform modular lattices and affine buildings

TL;DR

This paper establishes a lattice-theoretic characterization of affine buildings of type A by introducing uniform modular lattices, showing their equivalence with these buildings, paralleling known results for spherical buildings.

Contribution

It introduces uniform modular lattices and proves their equivalence with affine buildings of type A, providing a new lattice-theoretic perspective.

Findings

Uniform modular lattices are equivalent to affine buildings of type A.

The work provides an affine counterpart to the known correspondence between projective geometries and spherical 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.