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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2603.27545 |
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| _version_ | 1866914429897015296 |
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| author | Sakamoto, Ryotaro Suzuki, Miyu Tamori, Hiroyoshi |
| author_facet | Sakamoto, Ryotaro Suzuki, Miyu Tamori, Hiroyoshi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27545 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Root lattices over totally real fields Sakamoto, Ryotaro Suzuki, Miyu Tamori, Hiroyoshi Combinatorics Number Theory 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. |
| title | Root lattices over totally real fields |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2603.27545 |