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).

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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

• Leech lattice, E8 lattice

• Niemeier lattice

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.

Related concepts

• integral lattice

• modular integral lattice

• unimodular integral lattice

• torus, elliptic curve

• genus of a lattice

• crystallographic group

• Bravais lattice

• sphere packing

• lattice model

  • lattice gauge theory

References

• Peter Engel, Louis Michel, Marjorie Senechal, Lattice Geometry, IHES/P/04/45 (2004) $[$pdf, cds:859509$]$

• Andries E. Brouwer, Lattices, Course notes (2002) [pdf, pdf]

• John Conway, N. Sloane, Low dimensional lattices I: Quadratic forms of small determinant

• Wikipedia, Lattice (group)

• Wikipedia, Lattice (module)

Applications in heterotic string theory and F-theory:

• Dmitry Manning-Coe, Lattices: From Roots to String Compactifications [arXiv:2304.05394]