nLab
lattice (discrete subgroup)

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

Definition

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

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

Examples

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

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

References

Revised on February 18, 2018 12:20:37 by Todd Trimble (69.112.103.83)