Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Root lattices over totally real fields

Ryotaro Sakamoto, Miyu Suzuki, Hiroyoshi Tamori

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.

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.

Paper Structure

This paper contains 20 sections, 37 theorems, 76 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions of root lattices
  4. Finite Coxeter systems
  5. Structure of root systems
  6. Fundamental system of roots
  7. Irreducibility of root lattices
  8. Comparison with Coxeter systems
  9. The case of rank greater than $2$
  10. The case of type $A_n$
  11. The case of type $B_n$
  12. The case of type $D_n$
  13. The case of type $E_n$, $F_4$ and $H_n$
  14. Summary for the case of rank greater than $2$
  15. The case of rank $2$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $K$ be a totally real number field and $\mathcal{O}=\mathcal{O}_K$ its ring of integers. Suppose that $n$ is an integer greater than $2$. Then, there exists an irreducible root lattice over $\mathcal{O}$ of type $X_n$ if and only if $c(X_n)\in K$.

Theorems & Definitions (80)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4: Kronecker's theorem
  • proof
  • Corollary 2.5
  • proof
  • Lemma 2.6
  • ...and 70 more