Root lattices over totally real fields
Abstract
A root lattice is a finite rank -lattice generated by elements satisfying . It is well-known that the root lattices have an classification and they play a prominent role in the study of even unimodular lattices. The notion of root lattices can be naturally generalized to lattices over the ring of integers of a totally real field . In the case where is a real quadratic field, such lattices were classified by Mimura in 1979, and this classification has been used by several researchers in the study of even unimodular -lattices. In this paper, we extend this classification to arbitrary totally real fields. The irreducible root lattices of rank greater than are indexed by finite Coxeter systems. All the rank root lattices are realized as orders in quadratic extensions of and their classification requires some technique from algebraic number theory.
Keywords
Cite
@article{arxiv.2603.27545,
title = {Root lattices over totally real fields},
author = {Ryotaro Sakamoto and Miyu Suzuki and Hiroyoshi Tamori},
journal= {arXiv preprint arXiv:2603.27545},
year = {2026}
}