List of Geometric Lattices and Integral Forms

Introduction

This article lists the lattices of the corpus, their covolumes, their duals and the integral forms they carry, from the elementary lattice $\mathbb{Z}^n$ to the Lipschitz and Hurwitz lattices of the quaternion algebra. Every entry points to the article that introduces the lattice or the form, and the article introduces nothing and proves nothing.

A lattice is a free abelian group of finite rank with an embedding in a real vector space, and the data that classify it geometrically are its Gram matrix, its covolume and its dual with respect to a form. The list gathers the abstract lattices as free $\mathbb{Z}$-modules, the Euclidean lattices and their covolumes, the dual lattices and the discriminants, the integral quadratic and bilinear forms that a lattice carries, the lattices of the ring of integers of a number field, and the two quaternion orders of the corpus, the Lipschitz and the Hurwitz lattice. The lattice in the sense of a discrete subgroup of finite covolume in a Lie group is a different object, and it is recorded in the warnings with the article that treats it.

The article records examples and non-examples side by side. Beside the lattices it lists the dense subgroups that fail discreteness, the sets that are not finitely generated, the lattices that are not free, and the second sense in which the corpus uses the word lattice, each with the failure named and the article that records it.

Lattices as Free $\mathbb{Z}$-Modules

A lattice is a free abelian group of finite rank, and its rank, its sublattices and its indices are the algebraic data on which the geometry is built.

Object	The property it has	Introduced in
a lattice $\Lambda$	a free $\mathbb{Z}$-module of finite rank $n$	Lattices and the Quaternion Lattice
the rank $n$	the number of generators, the dimension of the ambient space	Lattices and the Quaternion Lattice
the change-of-basis matrix $A$	the matrix relating two bases, of determinant equal to the index	Lattices and the Quaternion Lattice
a sublattice $\Lambda' \leq \Lambda$	a free submodule of the same rank, of finite index in the geometric case	Lattices and the Quaternion Lattice
the index $[\Lambda:\Lambda']$	the determinant of the change of basis, the ratio of the covolumes	Lattices and the Quaternion Lattice
the free $\mathbb{Z}$-module of rank $n$	the abstract lattice, without an embedding	Direct Sums, Free Modules and Rank
the elementary lattice $\mathbb{Z}^n \leq \mathbb{R}^n$	the standard lattice, a rank-$n$ subgroup with covolume $1$	Lattices in Lie Groups
the period lattice $\Lambda = \mathbb{Z}\omega_1 + \mathbb{Z}\omega_2$	the rank-two lattice of the periods of the elliptic functions, on which $\mathbb{C}/\Lambda$ is an elliptic curve	Elliptic Functions and Integrals (planned)

The lattice is a free module, so it has a basis, and a sublattice of finite index has a basis whose change-of-basis matrix has determinant equal to the index. The elementary lattice and its images by invertible linear maps are the Euclidean lattices, and the abstract free module is the algebraic object from which they are obtained by a choice of embedding.

The Euclidean Lattice and its Covolume

An embedded lattice has a Gram matrix, and the square root of its determinant is the covolume, the volume of a fundamental domain.

Object	The property it has	Introduced in
the Gram matrix $G = (v_i\cdot v_j)$	the matrix of the inner products of a basis, encoding the metric of the lattice	Lattices and the Quaternion Lattice
the covolume $\operatorname{covol}(\Lambda) = \sqrt{\det G}$	the volume of a fundamental domain, the density of the embedding	Lattices and the Quaternion Lattice
the scaling law $\operatorname{covol}(\Lambda') = [\Lambda:\Lambda']\operatorname{covol}(\Lambda)$	the index times the covolume is the covolume	Lattices and the Quaternion Lattice
the lattice $A\mathbb{Z}^n$ in $\mathbb{R}^n$	a lattice of covolume $\lvert\det A\rvert$	Lattices in Lie Groups
the integral lattice	a lattice with a positive-definite integral Gram matrix	Lattices and the Quaternion Lattice
the covolume of the arithmetic quotient	the volume $\mu(G/\Gamma)$ of a lattice in a Lie group, defined up to normalisation	Lattices in Lie Groups

The Gram matrix is the complete metric data of the lattice, and the covolume is its geometric size: it is the volume of the parallelotope spanned by a basis, and it multiplies by the index under the passage to a sublattice. The covolume of a lattice in a Lie group is the volume of the quotient, and it is the invariant by which the arithmetic lattices are compared; the two covolumes share the name and the scaling property, though the objects are different.

The Dual Lattice and the Discriminant of the Form

The dual lattice is the set of vectors whose inner products with the lattice are integral, and the discriminant of the form together with the order of the quotient $\Lambda^*/\Lambda$ measures the embedding of the lattice in its dual.

Object	The property it has	Introduced in
the dual lattice $\Lambda^*$	the vectors $y$ with $x\cdot y \in \mathbb{Z}$ for all $x \in \Lambda$	Lattices and the Quaternion Lattice
the discriminant of the form	the determinant of the Gram matrix up to squares, an invariant of the isometry class	Bilinear Forms, §Congruence and the Discriminant
the integrality of the norm	the norm $N(x) = x x^{\natural}$ is integer-valued on the Lipschitz and the Hurwitz lattice	Lattices and the Quaternion Lattice
the discriminant $d_K$ of a number field	the determinant of an integral basis of $\mathcal{O}_K$	Algebraic Number Theory
the ring of integers $\mathcal{O}_K$	a lattice of rank $n = [K:\mathbb{Q}]$, with an integral basis	Algebraic Number Theory

The dual lattice sits between the lattice and its real span, the quotient $\Lambda^*/\Lambda$ is the finite group that measures the failure of the lattice to be self-dual, of order four for the Hurwitz lattice, For the ring of integers the discriminant is the arithmetic invariant of the number field, and the ramified primes are exactly its prime divisors.

Integral Forms

An integral form is a bilinear or quadratic form taking integral values on a lattice, and the form is the datum that makes the lattice a geometric object.

Object	The property it has	Introduced in
an integral quadratic form	a quadratic form $q$ with $q(x) \in \mathbb{Z}$ for all lattice vectors	Quadratic Forms and Polarisation
the polar form $g(u,v) = \tfrac12(q(u+v)-q(u)-q(v))$	the bilinear form associated with a quadratic form	Quadratic Forms and Polarisation
an integral bilinear form	a symmetric bilinear form with integral values on the lattice	Bilinear Forms
the Gram matrix of a form	the matrix of an integral form, an integral matrix	Bilinear Forms
the norm $N(x) = x x^{\natural}$	the quadratic form of the quaternion algebra, integral on the quaternion lattices	Lattices and the Quaternion Lattice
the polar form $(x,y) = \operatorname{Sc}(x y^{\natural})$	the Euclidean dot product, the polar form of the norm	Lattices and the Quaternion Lattice
the radical of an integral form	the vectors orthogonal to the whole space, the obstruction to nondegeneracy	Bilinear Forms, §Rank and the Radical

The passage from an integral form to a lattice and back is the arithmetic content of the geometry: the Gram matrix of a basis is the matrix of the form, the covolume is its determinant, the dual lattice is the lattice of the inverse form, and the integrality of the form is the condition $\Lambda\subseteq\Lambda^*$. The norm of the quaternion algebra is the classical example of an integral quaternary form, and its polar form is the Euclidean dot product.

The Quaternion Lattices

The Lipschitz and Hurwitz lattices are the integral forms of the quaternion algebra, and the Hurwitz lattice is the maximal order.

Object	The property it has	Introduced in
the Lipschitz lattice $\mathbb{H}(\mathbb{Z})$	the integer span of the quaternion basis, rank $4$, not maximal	Lattices and the Quaternion Lattice
the Hurwitz lattice $\mathbb{H}'(\mathbb{Z})$	the order with the half-integral generator $\omega = \tfrac12(1+e_1+e_2+e_3)$, rank $4$, maximal	Lattices and the Quaternion Lattice
the Hurwitz unit group	the $24$ units of the Hurwitz order, the binary tetrahedral group $2T$	Quaternion Geometry, §The Units and Their Geometry
the norm on the lattice	the sum of four squares, the integral quadratic form of the lattice	Quaternion Geometry, §The Arithmetic Geometry of the Quaternion Lattice
the unit sphere lattice	the unit sphere carries a discrete subgroup of finite covolume, up to scaling	Quaternion Geometry, §The Units and Their Geometry

The Hurwitz order contains the Lipschitz order with index two, and it is the maximal order of the rational quaternion algebra; the unit group is finite of order $24$ and is the binary tetrahedral group. The lattice is the arithmetic counterpart of the geometry of the quaternion sphere, and the norm is the quaternary quadratic form that the lattice carries.

Warnings

An object that a reader may expect among the geometric lattices and integral forms, and does not find, is recorded with the reason.

Object	Why it is not listed as a geometric lattice	Introduced in
a lattice in a Lie group	a discrete subgroup of finite covolume, a different sense of the word, treated with the discrete geometric groups	Lattices in Lie Groups
a non-uniform lattice	finite covolume but noncompact quotient, with cusps; it is a lattice in the Lie-group sense	Lattices in Lie Groups
a dense subgroup of $\mathbb{R}^n$	not discrete, so not a lattice in any sense	Topological Groups
the rational lattice $\mathbb{Q}^n$	not discrete and not finitely generated over $\mathbb{Z}$	Direct Sums, Free Modules and Rank
the dual of a lattice with no form fixed	the dual depends on the form; without one the construction is undefined	Lattices and the Quaternion Lattice

Summary

This article has listed the geometric lattices and integral forms of the corpus. The lattices as free $\mathbb{Z}$-modules open the list, with their rank, their sublattices and their indices; the Euclidean lattices follow, with the Gram matrix and the covolume, together with the covolume of a lattice in a Lie group; the dual lattice, the discriminant and the ring of integers of a number field are recorded; the integral quadratic and bilinear forms are gathered with the polar form, the norm and the radical; and the Lipschitz and Hurwitz lattices close the list with the norm and the Hurwitz unit group. Beside the examples stand the non-examples: a dense subgroup is not discrete, the rational lattice is not finitely generated, the dual is undefined without a form, and a lattice in a Lie group is a lattice in the other sense of the word.

Summary of Notation

A catalogue denotes its objects by name rather than by symbol. The symbols that appear in the tables are the following, and they are the symbols of the articles that introduce them.

Symbol	Meaning
$\Lambda$, $\Lambda^*$, $n$, $[\Lambda:\Lambda']$	Lattice, dual, rank, index
$G = (v_i\cdot v_j)$, $\operatorname{covol}(\Lambda) = \sqrt{\det G}$	Gram matrix and covolume
$\mathbb{Z}^n$, $A\mathbb{Z}^n$	Elementary lattice and its linear images
$\mathcal{O}_K$, $d_K$	Ring of integers and discriminant of a number field
$q$, $g(u,v)$, $N(x) = x x^{\natural}$	Quadratic form, polar form, quaternion norm
$\mathbb{H}(\mathbb{Z})$, $\mathbb{H}'(\mathbb{Z})$	Lipschitz lattice, Hurwitz lattice
$\omega = \tfrac12(1+e_1+e_2+e_3)$	Half-integral generator of the Hurwitz order
$\mu(G/\Gamma)$, $\operatorname{covol}(\Gamma)$	Haar measure and covolume of a lattice in a Lie group
$\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$	The standard number systems of the corpus
$\operatorname{Sc}(x)$	The scalar part of a quaternion product