---
tag: Z4FSES3CZ0
kind: definition
name: Lattice (group)
msc: [11H06]
defines: WN9V7WSJKF
uses: [0XPR1E85W0, 0X1JXT47GA, 0CRMGPAH83, 00KDZRW4C0]
sources:
  - url: https://en.wikipedia.org/wiki/Lattice_(group)
---

## Definition

Let $n \ge 1$ and consider the [[Euclidean space|0XPR1E85W0]] $\R^n$ as a group under addition.
A *lattice* in $\R^n$ is a [[subgroup|0X1JXT47GA]] of the form
$$\Lambda = \Z v_1 + \Z v_2 + \dots + \Z v_n = \set{k_1 v_1 + \dots + k_n v_n : k_1, \dots, k_n \in \Z},$$
where $v_1, \dots, v_n$ are [[linearly independent|0CRMGPAH83]] vectors of $\R^n$.
The family $(v_1, \dots, v_n)$ is a *basis* of $\Lambda$.
The *covolume* (or *determinant*) of $\Lambda$ is $\det \Lambda = \abs{\det(v_1, \dots, v_n)}$.

## Remarks

A subgroup $\Lambda$ of $\R^n$ is a lattice if and only if it is discrete and spans $\R^n$ as a real vector space.
Two bases of the same lattice differ by an integer matrix of [[determinant|00KDZRW4C0]] $\pm 1$, so the covolume does not depend on the basis.
It equals the volume of the fundamental parallelepiped $\set{t_1 v_1 + \dots + t_n v_n : 0 \le t_i < 1}$.

Example: $\Z^n$ is a lattice with basis the standard vectors $e_1, \dots, e_n$ and covolume $1$.

Non-example: $\Z + \Z\sqrt{2} \subset \R$ is a subgroup generated by two vectors, but two vectors in $\R^1$ are never linearly independent.
In fact this subgroup is dense in $\R$, so it is not discrete.
Another non-example: $\Z \times \set{0}$ in $\R^2$ is discrete but does not span $\R^2$.