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arXiv · 2603.27545

Root lattices over totally real fields

Ryotaro Sakamoto·Miyu Suzuki·Hiroyoshi Tamori
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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.

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Ryotaro Sakamoto, Miyu Suzuki, Hiroyoshi Tamori. 2026-03-29. Root lattices over totally real fields. https://arxiv.org/abs/2603.27545

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