List of Geometric Forms

Introduction

This article lists the forms of the corpus — the bilinear, quadratic, sesquilinear, Hermitian, alternating and symplectic forms — each with the space it is defined on, the invariants that classify it and the article that introduces it. A form is a scalar-valued rule on two vectors, or on one vector, that is linear or conjugate-linear in its arguments; the structure it defines on the space is what the geometric groups of the following catalogues are required to preserve. The list records the invariants — the rank, the radical, the discriminant, the signature, the Witt index — and the classification each classical field affords.

Every entry points to the article that introduces the form or the invariant. The article introduces nothing and proves nothing: it records the classifying data the introducing article establishes, and it neither restates a definition nor gives a proof.

The article records examples and non-examples side by side. Beside the forms that the classification separates it lists the cases in which the classification collapses or the correspondence fails: the quadratic forms and the symmetric bilinear forms, which are in bijection only when $2$ is invertible in the base ring; the alternating forms, which are not the skew-symmetric forms in characteristic $2$; the split octonions, whose norm is isotropic although the octonion norm of the division algebra is not; and the forms over $\mathbb{Q}$ and over a number field, whose classification needs the local invariants and is not read from the signature alone, each with the failure named and the article that records it.

The Bilinear and Sesquilinear Forms

The bilinear form is the basic datum: a rule linear in each argument, represented by a Gram matrix once a basis is chosen, with the radical measuring its degeneracy. The sesquilinear and Hermitian forms are the case of a non-commutative ring with an involution.

Form or invariant	Its classifying data	Introduced in
a bilinear form $B : M \times M \to R$	bilinearity; the Gram matrix $G_{ij} = B(e_i,e_j)$ in a basis, with $G \mapsto P^TGP$ under a change of basis	Bilinear Forms
the radical $\operatorname{rad}(B)$	$\{u : B(u,v) = 0 \ \forall v\}$; the form is non-degenerate when the radical is zero	Bilinear Forms
the rank $\operatorname{rank}(B)$	the rank of the Gram matrix; the discriminant $\Delta(B) = \det G \in F^\times/(F^\times)^2$	Bilinear Forms
the orthogonal direct sum $B_1 \perp B_2$	the form on $M_1 \oplus M_2$ with $B(M_1,M_2) = 0$; the operation of the classification	Bilinear Forms
the isometry group $\operatorname{Isom}(M,B)$	the linear maps preserving $B$; $\operatorname{O}(M,B)$ in the symmetric case, $\operatorname{Sp}(M,B)$ in the alternating case	Bilinear Forms; List of Classical Geometric Groups
a $\sigma$-sesquilinear form	linear in one argument and $\sigma$-linear in the other for an involution $\sigma$ of the ring	Hilbert Algebras
a Hermitian form $s(x,y)$	$s(x,y) = \sigma(s(y,x))$; the diagonal $q(x) = s(x,x)$ is a quadratic form over the fixed ring	Hilbert Algebras
the Hermitian Gram matrix $H$	$H^\dagger = \sigma(H)^T = H$; the unitary group is its isometry group	Hilbert Algebras; The Unitary and Symplectic Groups
the trace form $T(x,y) = \operatorname{Tr}(m_{xy})$	the regular trace of a finite-dimensional algebra; the reduced trace form in the central simple case	Hilbert Algebras
the reduced norm $\operatorname{Nrd}$	the norm of a central simple algebra of degree $d$; a form of dimension $d^2$	Hilbert Algebras

The Quadratic Forms

A quadratic form is homogeneous of degree two, and it is recovered from its polar form when $2$ is invertible; its classification over the classical fields is the content of Sylvester's law and of the rank.

Form or invariant	Its classifying data	Introduced in
a quadratic form $q : M \to R$	$q(av) = a^2q(v)$ and the polar identity; the polar form $B$ with $q(v) = B(v,v)$	Quadratic Forms and Polarisation
the passage to the polar form	a bijection between quadratic and symmetric bilinear forms when $2$ is invertible; a genuine obstruction otherwise	Quadratic Forms and Polarisation
the diagonal form $\langle a_1,\ldots,a_n\rangle$	$\sum_i a_ix_i^2$; the diagonalisation of a form over a field of characteristic not $2$	Quadratic Forms and Polarisation
Sylvester's law of inertia	over $\mathbb{R}$ a form is classified by its signature $(p,r)$; the indices are invariant	Quadratic Forms and Polarisation
the signature $\sigma(q) = p - r$ and the rank $p + r$	the real classification; the complex classification is the rank alone	Quadratic Forms and Polarisation; Pseudo-Riemannian and Lorentzian Geometry
the discriminant $\Delta(q)$	$a_1\cdots a_n \in F^\times/(F^\times)^2$; an invariant of the isometry class	Quadratic Forms and Polarisation; Bilinear Forms
the hyperbolicity and the isotropic vectors	a form is isotropic when $q(v) = 0$ for some $v \neq 0$; the hyperbolic plane is the model	Witt's Theorems
the norms of the algebras	$N(x) = x\bar x$ on $\mathbb{C}$, $\mathbb{H}$, $\mathbb{B}$ and $\mathbb{O}$; a composition law $N(xy) = N(x)N(y)$ only in dimensions $1$, $2$, $4$, $8$	Quadratic Forms over Algebras and Norms
a Pfister form $\langle\!\langle a_1,\ldots,a_n\rangle\!\rangle$	the norm of a composition algebra of dimension $2^n$; the tensor product $\bigotimes_i\langle 1,-a_i\rangle$	Quadratic Forms over Algebras and Norms; The Witt Group and the Grothendieck–Witt Ring

The Alternating and Symplectic Forms

An alternating form satisfies $\omega(u,u) = 0$, which over a field of characteristic not $2$ is the skew-symmetry of its matrix; the non-degenerate alternating forms are the symplectic forms, and their normal form is the linear Darboux theorem.

Form or invariant	Its classifying data	Introduced in
an alternating form $\omega$	$\omega(u,u) = 0$ for all $u$; equivalently $\omega(u,v) = -\omega(v,u)$ when $2$ is invertible	Symplectic Forms and Poisson Brackets; Bilinear Forms
an alternating $2$-form as a wedge	an element of $\Lambda^2(V^)$; the algebra of alternating forms is $\Lambda(V^)$	The Determinant and Alternating Forms
the Pfaffian	$\operatorname{Pf}(\Omega)$ for an alternating matrix of even size; $\det\Omega = \operatorname{Pf}(\Omega)^2$, so a symplectic map has determinant $1$	The Determinant and Alternating Forms; Symplectic Forms and Poisson Brackets
a symplectic form	a non-degenerate alternating form on a finite-dimensional space; the dimension is even	Symplectic Forms and Poisson Brackets
the linear Darboux normal form	the symplectic basis $e_1,\ldots,e_n,f_1,\ldots,f_n$ with $\omega(e_i,f_j) = \delta_{ij}$; $\omega_0$ has matrix $J$	Symplectic Forms and Poisson Brackets
the symplectic group $Sp(V,\omega)$	the isometries of $\omega$; $A^TJA = J$; contained in $SL(V)$ by the Pfaffian	Symplectic Forms and Poisson Brackets; The Unitary and Symplectic Groups
the Lagrangians	the maximal totally isotropic subspaces; the Lagrangian Grassmannian	Symplectic Forms and Poisson Brackets; Symplectic Geometry

The Classification Theorems and Their Invariants

The classification of quadratic forms over a field is governed by Witt's theorems; over $\mathbb{R}$, $\mathbb{C}$, $\mathbb{F}_q$ and $\mathbb{Q}$ it takes the forms recorded below, and the Witt ring is the algebraic home of the invariants.

Theorem or invariant	Its statement or classifying data	Introduced in
the Witt extension theorem	every isometry between subspaces of a non-degenerate quadratic space extends to the whole space; complements exist	Witt's Theorems
the Witt index $m$	the common dimension of the maximal totally isotropic subspaces; $V \cong V_0 \perp mH$ with $V_0$ anisotropic	Witt's Theorems
the Witt cancellation theorem	$q \perp q_1 \cong q \perp q_2 \Rightarrow q_1 \cong q_2$ for non-degenerate forms; the basis of the Witt ring	Witt's Theorems
the classification over $\mathbb{R}$	the signature $(p,r)$ by Sylvester's law; rank and signature	Quadratic Forms and Polarisation
the classification over $\mathbb{C}$	the rank alone; every non-degenerate complex form is $\langle 1,\ldots,1\rangle$	Quadratic Forms and Polarisation; Witt's Theorems
the classification over $\mathbb{F}_q$, $q$ odd	the dimension and the discriminant; the number of maximal totally isotropic subspaces of $mH$ is $\prod_{i=0}^{m-1}(q^i+1)$	Witt's Theorems
the Hasse–Minkowski principle	the classification over a number field by the completions; the local invariants determine the global form	Witt's Theorems
the Witt ring $W(F)$	the isometry classes modulo the hyperbolic forms, under $\perp$ and $\otimes$; the Grothendieck–Witt ring $GW(F)$ above it	The Witt Group and the Grothendieck–Witt Ring
the invariants $\dim$, $\Delta$, $\operatorname{sign}$	the dimension, the discriminant and the total signature $W(F) \to \prod_P\mathbb{Z}$	The Witt Group and the Grothendieck–Witt Ring
the Hasse invariant $e_2$	the invariant $I^2/I^3 \hookrightarrow \operatorname{Br}_2(F)$; the Milnor K-theory quotients $I^n/I^{n+1}$	The Witt Group and the Grothendieck–Witt Ring

The Forms on the Number Systems

The norms of the number systems are the examples through which the general theory is read, and the Cayley–Dickson doubling produces them in dimensions one, two, four and eight.

Space and form	The form and its properties	Introduced in
$\mathbb{C}$ with $N(z) = z\bar z$	the positive definite norm of dimension $2$; a composition law	Quadratic Forms over Algebras and Norms; The Complex Numbers
$\mathbb{H}$ with $N(q) = q\bar q$	the positive definite quaternion norm of dimension $4$; a composition law; anisotropic	Quadratic Forms over Algebras and Norms; Quaternion Algebra
$\mathbb{B}$ with $N(q) = q\bar q$	the complex-valued biquaternion norm; the real part and imaginary part are forms of signature $(4,4)$	Quadratic Forms over Algebras and Norms; The Clifford Structure of the Biquaternion Algebra
the split biquaternions $\mathbb{H}_{\mathbb{D}}$	a norm that is isotropic; the split form of the quaternion norm	Quadratic Forms over Algebras and Norms; Split-Biquaternions and Hyperbolic Geometry
$\mathbb{O}$ with $N(x) = x\bar x$	the octonion norm of dimension $8$; a composition law; anisotropic	Quadratic Forms over Algebras and Norms; Octonion Algebra
the split octonions	the isotropic norm of dimension $8$; the $\mu = +1$ double of $\mathbb{H}_{\mathbb{D}}$	Quadratic Forms over Algebras and Norms
the quadric $\mathcal{Q}(Q)$	the projective quadric $\{[x] : Q(x) = 0\}$ of an $A$-valued form	Quadratic Forms over Algebras and Norms; Projective Geometry

Non-examples and Warnings

Object	Why the expected statement fails	Introduced in
a quadratic form over $\mathbb{F}_2$	the polar form does not determine the quadratic form when $2$ is not invertible; the correspondence is an obstruction, not a bijection	Quadratic Forms and Polarisation
an alternating form in characteristic $2$	it is not the same as a skew-symmetric form; skew-symmetry with $\omega(u,u) = 0$ is the condition available over every ring	The Determinant and Alternating Forms; Symplectic Forms and Poisson Brackets
the split octonions	their norm is isotropic, unlike the norm of the division algebra $\mathbb{O}$; the two are different forms of the same dimension	Quadratic Forms over Algebras and Norms
the classification over $\mathbb{Q}$	it is not read from the signature alone; the Hasse–Minkowski principle needs the local invariants at the completions	Witt's Theorems
the Hermitian form over a non-commutative ring	the diagonal $q(x) = s(x,x)$ is a quadratic form over the fixed ring and not over the base; the transfer is part of the theory	Hilbert Algebras
a degenerate form, $\langle 0\rangle$	its matrix is not invertible and there is no discriminant in $F^\times/(F^\times)^2$; the radical is the whole space	Bilinear Forms
a form of odd dimension over a finite field	the discriminant alone does not classify it in characteristic $2$; the classification stated above needs $q$ odd	Witt's Theorems

Objects that a reader may expect in a list of geometric forms, and does not find here.

Object	Why it is not listed	Introduced in
differential forms	they are forms on the tangent bundle of a manifold, a Part III construction, and are catalogued with the manifolds	Differential Forms and Stokes' Theorem
the integral forms of a lattice	they are recorded with the geometric lattices and the integral quadratic forms	List of Geometric Lattices and Integral Forms
the symplectic form of a symplectic manifold	it is a closed non-degenerate $2$-form on a manifold; the algebraic form is listed here and the manifold there	Symplectic Geometry; Symplectic and Contact Topology
the Killing form of a Lie algebra	an invariant symmetric bilinear form on a Lie algebra, recorded with the structure theory	Structure of Lie Algebras
the Petersson inner product and the automorphic forms	the Hermitian forms of the analytic theory belong to a different catalogue	Automorphic Forms

Summary

This article has listed the forms of the corpus: the bilinear forms with their Gram matrices, radicals and discriminants; the sesquilinear and Hermitian forms with an involution, their trace forms and reduced norms; the quadratic forms with their polar forms, diagonalisations, signatures and discriminants, and the norms of $\mathbb{C}$, $\mathbb{H}$, $\mathbb{B}$, the split biquaternions and $\mathbb{O}$; the alternating and symplectic forms with the Pfaffian, the Darboux normal form and the Lagrangians; and the classification theorems — Witt's extension, index and cancellation, Sylvester's law, the classifications over $\mathbb{C}$ and $\mathbb{F}_q$, the Hasse–Minkowski principle, and the Witt and Grothendieck–Witt rings with their invariants. Beside the examples stand the non-examples: the characteristic-$2$ failures of the polar correspondence and of alternation, the isotropic split octonion norm, the forms over $\mathbb{Q}$ that need their local invariants, and the degenerate forms without a discriminant. The list introduces and proves nothing; it is the index of the forms of the corpus.

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
$B$	a bilinear form, symmetric unless stated
$G = (G_{ij})$	the Gram matrix of a bilinear form in a basis
$\operatorname{rad}(B)$, $\operatorname{rank}(B)$	the radical and the rank of a form
$\Delta$	the discriminant, in $F^\times/(F^\times)^2$
$\perp$	the orthogonal direct sum
$q$	a quadratic form; $B$ its polar form, $q(v) = B(v,v)$
$\langle a_1,\ldots,a_n\rangle$	the diagonal form $\sum_i a_ix_i^2$
$\sigma(q) = p - r$	the signature of a real quadratic form, of positive index $p$ and negative index $r$
$H$	the hyperbolic plane
$m$	the Witt index
$\omega$, $\omega_0$, $J$	an alternating $2$-form, the standard symplectic form, its matrix
$\operatorname{Pf}$	the Pfaffian
$Sp(V,\omega)$	the symplectic group
$W(F)$, $GW(F)$, $I$	the Witt ring, the Grothendieck–Witt ring, the fundamental ideal
$\langle\!\langle a_1,\ldots,a_n\rangle\!\rangle$	a Pfister form
$N$, $\operatorname{Nrd}$, $\operatorname{Tr}$, $\operatorname{Trd}$	a norm, the reduced norm, the trace and reduced trace
$s(x,y)$, $\sigma$, $A^\sigma$	a sesquilinear form, an involution, its fixed ring