# nLab lattice (discrete subgroup)

Contents

This entry is about the notion of lattice in group theory/quadratic form-theory. For other notions see at lattice (disambiguation).

# Contents

## Idea

Classically, a lattice in the Cartesian space $\mathbb{R}^n$ is a discrete subgroup (of the underlying topological abelian group) that spans $\mathbb{R}^n$ as a vector space over $\mathbb{R}$. This may be generalized, from $\mathbb{R}^n$ to a general locally compact abelian group.

## Definition

A lattice in a locally compact Hausdorff abelian group $A$ is a subgroup $L \hookrightarrow A$ that is discrete and cocompact, meaning that the quotient group $A/L$ with the quotient topology is compact.

Applying Pontryagin duality, the dual of the quotient map $q: A \to A/L$ is in that case a discrete subgroup $\widehat{A/L} \hookrightarrow \widehat{A}$ that is also cocompact (its cokernel being the compact group $\widehat{L}$). This is called the dual lattice of $L$.

## Examples

Notable examples of classical lattices (in $\mathbb{R}^n$) include

The standard diagonal inclusion of a global field (such as a number field) $k$ into its ring of adeles $A_k$ is a lattice in the more general sense. Recalling that $A_k$ is Pontryagin dual to itself, the lattice $k$ is identified with its dual lattice.

## References

Last revised on June 14, 2022 at 07:07:16. See the history of this page for a list of all contributions to it.