Root lattices over totally real fields

TL;DR

This paper extends the classification of root lattices over totally real fields beyond quadratic cases, linking them to Coxeter systems and algebraic number theory techniques.

Contribution

It generalizes the classification of root lattices over totally real fields, previously known only for quadratic fields, to arbitrary totally real fields.

Findings

Rank > 2 irreducible root lattices are indexed by Coxeter systems.

Rank 2 root lattices are realized as orders in quadratic extensions.

The classification involves algebraic number theory techniques.

Abstract

A root lattice is a finite rank $\mathbb{Z}$-lattice generated by elements $x$ satisfying $x\cdot x=2$. It is well-known that the root lattices have an $ADE$ 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 $\mathcal{O}$ of a totally real field $K$. In the case where $K$ 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 $\mathcal{O}$-lattices. In this paper, we extend this classification to arbitrary totally real fields. The irreducible root lattices of rank greater than $2$ are indexed by finite Coxeter systems. All the rank $2$ root lattices are realized as orders in quadratic extensions of $K$ and their classification requires some technique from algebraic number theory.