English

Root lattices over totally real fields

Combinatorics 2026-03-31 v1 Number Theory

Abstract

A root lattice is a finite rank Z\mathbb{Z}-lattice generated by elements xx satisfying xx=2x\cdot x=2. It is well-known that the root lattices have an ADEADE 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 O\mathcal{O} of a totally real field KK. In the case where KK 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 O\mathcal{O}-lattices. In this paper, we extend this classification to arbitrary totally real fields. The irreducible root lattices of rank greater than 22 are indexed by finite Coxeter systems. All the rank 22 root lattices are realized as orders in quadratic extensions of KK 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}
}
R2 v1 2026-07-01T11:42:41.772Z