List of Classical Geometric Groups

Introduction

This article lists the classical geometric groups of the corpus — the orthogonal, unitary and symplectic groups — each with the form it preserves and the article that defines it. A classical group is the isometry group of a non-degenerate form: a symmetric bilinear form gives the orthogonal group $\operatorname{O}(V,q)$ and its determinant-one subgroup $\operatorname{SO}(V,q)$, a Hermitian form gives the unitary group $\operatorname{U}(V,h)$ and $\operatorname{SU}(V,h)$, and an alternating form gives the symplectic group $\operatorname{Sp}(V,\omega)$. With the general and special linear groups of the preceding catalogue, these are the groups that the corpus uses to generate the four infinite families $A$, $B$, $C$, $D$ of the classification of simple Lie algebras.

Every entry points to the article that introduces the group. The article introduces nothing and proves nothing: it records the form each group preserves, the determinant constraint, the centre, the compactness and the family, and it neither restates a definition nor gives a proof.

The article records examples and non-examples side by side. Beside the groups it lists the cases in which an expected parallel fails or a symbol carries two meanings: the orthogonal group is not connected, so it fails the connectedness that the other compact classical groups have; $\operatorname{SO}(n)$ has centre $\{\pm I\}$ for even $n$ and trivial centre for odd $n$, so the centre is not uniform in the family; the symbol $\operatorname{Sp}$ denotes both the compact quaternionic unitary group $Sp(n)$ and the split symplectic group $Sp(2n,F)$; the quaternionic determinant cannot be defined as an ordinary determinant and requires the Dieudonné construction; a degenerate form has an isometry group that is not a classical group; and $\operatorname{O}(V,q)$ is defined by a form, so it is not a group-layer object, each with the failure named and the article that records it.

The Construction from a Form

The three classical families are the isometry groups of the three types of non-degenerate form, and the definition is uniform: an isometry is a linear isomorphism preserving the form.

Object	The property it has	Introduced in
the isometry group $\operatorname{Isom}(V,\beta)$	$\{g \in GL(V) : \beta(gx,gy) = \beta(x,y)\}$; the definition of a classical group from a form	Matrix Groups and Classical Groups; Bilinear Forms
the orthogonal group $\operatorname{O}(V,q)$	the isometries of a quadratic form $q$, equivalently of its polar form $B$	Isometries and Orthogonal Transformations; The Unitary and Symplectic Groups
the special orthogonal group $\operatorname{SO}(V,q)$	the isometries of determinant $+1$; the kernel of $\det$ on $\operatorname{O}(V,q)$	Isometries and Orthogonal Transformations; The Rotation Group and Orientation
the similarity group $GO(V,q)$	the maps with $q(Tv) = c\,q(v)$ for a multiplier $c$; its multiplier group	Isometries and Orthogonal Transformations
the unitary group $\operatorname{U}(V,h)$	the isometries of a Hermitian form; a sesquilinear form with an involution	The Unitary and Symplectic Groups; Hilbert Algebras
the special unitary group $\operatorname{SU}(V,h)$	the isometries of determinant one; $\operatorname{SU}(V,h) = \operatorname{U}(V,h) \cap SL(V)$	The Unitary and Symplectic Groups
the symplectic group $\operatorname{Sp}(V,\omega)$	the isometries of a non-degenerate alternating form; contained in $SL(V)$ by the Pfaffian	The Unitary and Symplectic Groups; Symplectic Forms and Poisson Brackets
the family of a classical group	the orthogonal, unitary and symplectic groups generate the families $B_n$, $D_n$, $A_n$ and $C_n$	The Unitary and Symplectic Groups; Root Systems and Classification
the compact and split real forms	each complex classical group has a compact real form and split real forms; the forms are the $O(p,r)$ and $U(p,r)$	The Unitary and Symplectic Groups; The Orthogonal Lie Algebra

The Orthogonal Groups

The orthogonal group is generated by the reflections, and the Cartan–Dieudonné theorem controls the parity of the number of reflections; the determinant separates the two components of the real group.

Object or statement	The property it has	Introduced in
the reflection $\tau_v$	the isometry fixing the hyperplane $v^\perp$; $\tau_v(x) = x - 2B(x,v)v/q(v)$	Isometries and Orthogonal Transformations
the Cartan–Dieudonné theorem	every isometry of a non-degenerate quadratic space is a product of reflections, at most $n$ of them, and at most $n+1$ in the affine case	Isometries and Orthogonal Transformations; List of Affine and Euclidean Groups
the parity of the reflection length	the parity is an invariant; $\operatorname{SO}(V,q)$ is the even part	The Rotation Group and Orientation
the range of the determinant	$\det A = \pm 1$ for $A \in O(n)$; $\operatorname{O}(V,q)$ has at most two components over a field	Matrix Groups and Classical Groups
the orthogonal group $O(p,r)$	the isometry group of a real form of signature $(p,r)$; $\operatorname{O}(n) = O(n,0)$ compact, $\operatorname{O}(n,1)$ the Lorentz-type group	The Rotation Group and Orientation; Pseudo-Riemannian and Lorentzian Geometry
the centres	$Z(SO(n)) = \{\pm I\}$ for even $n \geq 4$ and trivial for odd $n \geq 3$; $Z(O(n)) = \{\pm I\}$	Matrix Groups and Classical Groups
the fundamental group of $SO(n)$	$\pi_1(SO(n)) = \mathbb{Z}/2$ for $n \geq 3$; the double cover is $\operatorname{Spin}(n)$	Matrix Groups and Classical Groups; List of Clifford Algebras and Spin Groups
the rotations of the plane and of space	$\operatorname{SO}(2) \cong S^1$; a rotation of $\mathbb{R}^3$ has an axis and an angle; $\operatorname{SO}(3) \cong \mathbb{RP}^3$	The Rotation Group and Orientation; Quaternion Rotations and Reflections

The Unitary Groups

A Hermitian form has a signature by the inertia theorem, and its isometry group is the unitary group of that signature; the definite case gives the compact group $U(n)$.

Object or statement	The property it has	Introduced in
the Hermitian Gram matrix $H$	$H^\dagger = H$; the form is $h(u,v) = u^\dagger Hv$	Hilbert Algebras; The Unitary and Symplectic Groups
the inertia theorem for Hermitian forms	a unique signature $(p,r,z)$ with normal form $\operatorname{diag}(I_p,-I_r,0_z)$; non-degenerate when $z = 0$	The Unitary and Symplectic Groups
the group $U(p,r)$	the isometry group of a Hermitian form of signature $(p,r)$, of real dimension $n^2$	The Unitary and Symplectic Groups
the group $SU(p,r)$	the determinant-one subgroup, of real dimension $n^2 - 1$	The Unitary and Symplectic Groups
the standard compact groups $U(n)$, $SU(n)$	$A^*A = I$ and $\det A = 1$; dimensions $n^2$ and $n^2 - 1$; $U(n)$ compact connected with centre $U(1)$	Matrix Groups and Classical Groups; The Unitary and Symplectic Groups
the sequence $1 \to SU(n) \to U(n) \to U(1) \to 1$	the determinant on $U(n)$ surjects onto the circle	Matrix Groups and Classical Groups
the intersection formula	$h = s + ia$ gives $\operatorname{U}(V,h) = \operatorname{O}(V_{\mathbb{R}},s) \cap \operatorname{Sp}(V_{\mathbb{R}},a)$	The Unitary and Symplectic Groups
the case $U(1,1)$	non-compact, containing a hyperbolic and a compact one-parameter subgroup; $SU(1,1) \cong SL(2,\mathbb{R}) \cong Sp(2,\mathbb{R})$	The Unitary and Symplectic Groups
the compact form inside the split form	$Sp(2m,\mathbb{R}) \cap O(2m) = U(m)$, a maximal compact subgroup	The Unitary and Symplectic Groups

The Symplectic Groups

An alternating form has even dimension over a field of characteristic not $2$, and its isometry group has determinant $1$ by the Pfaffian; the compact quaternionic unitary group is the compact real form of the family.

Object or statement	The property it has	Introduced in
the symplectic group $Sp(2m,F)$	$M^T\Omega M = \Omega$; dimension $2m^2 + m$; contained in $SL(2m,F)$	The Unitary and Symplectic Groups
the determinant constraint	every symplectic matrix has determinant $1$, by the Pfaffian	Symplectic Forms and Poisson Brackets; The Determinant and Alternating Forms
the case $Sp(2,F) = SL(2,F)$	the two-dimensional coincidence of the symplectic and special linear groups	The Unitary and Symplectic Groups
the compact quaternionic group $Sp(n)$	$A^*A = I$ over $\mathbb{H}$; compact connected; dimension $n(2n+1)$; the compact real form of $Sp(2n,\mathbb{C})$	Matrix Groups and Classical Groups; The Unitary and Symplectic Groups
the two uses of the symbol $\operatorname{Sp}$	$Sp(n)$ the compact quaternionic unitary group, $Sp(2n,F)$ the split symplectic group; separated only by context	Matrix Groups and Classical Groups; The Unitary and Symplectic Groups
the symplectic group of a form $\operatorname{Sp}(V,\omega)$	the isometries of a non-degenerate alternating form; the Lagrangians are its maximal isotropic subspaces	Symplectic Forms and Poisson Brackets
the quasi-split forms	$\operatorname{Sp}(V,\omega)$ has no non-trivial compact form other than $Sp(n)$; the split form is $Sp(2n,\mathbb{R})$	The Unitary and Symplectic Groups

The Four Families and the Small Coincidences

Over $\mathbb{C}$ the Lie algebras of the classical groups are the four infinite families, and the low-dimensional coincidences among the small groups are recorded with them.

Family	The group over $\mathbb{C}$	Rank and dimension	Introduced in
$A_n$	$SL(n+1,\mathbb{C})$, with compact form $SU(n+1)$	rank $n$, dimension $n(n+2)$	The Unitary and Symplectic Groups; Root Systems and Classification
$B_n$	$SO(2n+1,\mathbb{C})$, with compact form $SO(2n+1)$	rank $n$, dimension $n(2n+1)$	The Unitary and Symplectic Groups; The Orthogonal Lie Algebra
$C_n$	$Sp(2n,\mathbb{C})$, with compact form $Sp(n)$	rank $n$, dimension $n(2n+1)$	The Unitary and Symplectic Groups; Root Systems and Classification
$D_n$	$SO(2n,\mathbb{C})$, with compact form $SO(2n)$	rank $n$, dimension $n(2n-1)$	The Unitary and Symplectic Groups; The Orthogonal Lie Algebra
the duality of $B$ and $C$	equal dimension and rank, with dual root systems, long and short roots interchanged	—	The Unitary and Symplectic Groups
the exception $D_2$	$\mathrm{SO}(4) \cong \mathrm{SL}(2) \oplus \mathrm{SL}(2)$; $D_2$ is not simple, $D_n$ is simple for $n \geq 3$	—	The Unitary and Symplectic Groups
the coincidence $A_1 = B_1 = C_1$	$SL(2)$, $SO(3)$ and $Sp(2)$ share the Lie algebra $\mathrm{SL}(2)$	—	The Unitary and Symplectic Groups; The Orthogonal Lie Algebra
$SU(2)$ and the unit quaternions	$SU(2) \cong \mathbb{H}^1 = S^3 = Sp(1)$	—	Matrix Groups and Classical Groups
the double covers	$SU(2) \to SO(3)$ and $SU(2) \times SU(2) \to SO(4)$, with kernels $\{\pm1\}$ and $\{\pm(1,1)\}$	—	Matrix Groups and Classical Groups; Quaternion Rotations and Reflections

Non-examples and Warnings

Object	Why the expected statement fails	Introduced in
the orthogonal group $O(n)$	it is not connected; it has two components, and $\operatorname{SO}(n)$ is only one of them	Matrix Groups and Classical Groups
the centre of $\operatorname{SO}(n)$	it is $\{\pm I\}$ for even $n \geq 4$ and trivial for odd $n \geq 3$; the centre is not uniform in the family	Matrix Groups and Classical Groups
the quaternionic determinant	it cannot be an ordinary determinant and requires the Dieudonné construction, valued in $\mathbb{H}^\times/[\mathbb{H}^\times,\mathbb{H}^\times]$	Matrix Groups and Classical Groups
the isometry group of a degenerate form	it is not a classical group; non-degeneracy is part of the definition of $O$, $U$ and $Sp$	Bilinear Forms; The Unitary and Symplectic Groups
the compact unitary group $U(n)$	it is the quotient $(\operatorname{SU}(n) \times U(1))/\mu_n$ and not the direct product $\operatorname{SU}(n) \times U(1)$; the determinant sequence $1 \to \operatorname{SU}(n) \to U(n) \to U(1) \to 1$ has kernel $\mu_n$	Matrix Groups and Classical Groups
the group $Sp(n)$ of the quaternionic form	it is not the split group $Sp(2n,F)$, despite the shared symbol; the context separates them and the two have different real and complex dimensions	The Unitary and Symplectic Groups
the group $O(V,q)$ as a group-layer object	it is defined by a form; the group-layer transformation group is $\operatorname{GL}(V)$, and $O(V,q)$ belongs to the form-preserving layer	Transformation Groups; Matrix Groups and Classical Groups

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

Object	Why it is not listed	Introduced in
the general and special linear groups	they preserve the linear structure and no form; they are the linear geometric groups	List of Linear Geometric Groups
the projective quotients $PO$, $PSO$, $PU$, $PSU$, $PSp$	they are the quotients by the centre and are catalogued with the projective groups	List of Projective Geometric Groups
the spin and pin groups	they are constructed from the Clifford algebra and are listed with it	List of Clifford Algebras and Spin Groups
$\operatorname{GL}_n(\mathbb{H})$ and the quaternionic classical groups	the quaternionic linear group is recorded with the automorphism and matrix groups	Matrix Groups and Classical Groups
the finite classical groups $PSL_n(q)$, $PSp_{2n}(q)$, $PSU_n(q)$, $P\Omega^\pm_{2n}(q)$	the finite versions over $\mathbb{F}_q$ are recorded with the linear groups and the finite groups of Lie type	Finite Simple Groups of Lie Type; List of Linear Geometric Groups

Summary

This article has listed the classical geometric groups of the corpus: the orthogonal groups $\operatorname{O}(V,q)$ and $\operatorname{SO}(V,q)$, generated by reflections with the Cartan–Dieudonné theorem and the parity of the reflection length, together with the real forms $O(p,r)$; the unitary groups $\operatorname{U}(V,h)$ and $\operatorname{SU}(V,h)$, with the inertia theorem, the signature $(p,r,z)$, the groups $U(p,r)$ and the intersection formula with the orthogonal and symplectic groups; the symplectic groups $\operatorname{Sp}(V,\omega)$ and $Sp(2m,F)$, with the Pfaffian determinant constraint and the compact quaternionic group $Sp(n)$; and the four families $A$, $B$, $C$, $D$ with their ranks, dimensions and small coincidences. Beside the examples stand the non-examples: the disconnectedness of $O(n)$, the centre of $SO(n)$ that changes with the parity of $n$, the Dieudonné determinant, the degenerate forms and the two meanings of $\operatorname{Sp}$. The list introduces and proves nothing; it is the index of the classical groups 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 those of the articles that introduce them.

Symbol	Meaning
$V$, $q$, $B$	a finite-dimensional space, a quadratic form, its polar form with $q(v) = B(v,v)$
$h$, $s$, $a$, $\sigma$	a Hermitian form, its symmetric and alternating parts, the involution
$\omega$, $\Omega$	an alternating form, the standard alternating matrix
$\operatorname{O}(V,q)$, $\operatorname{SO}(V,q)$, $O(p,r)$	the orthogonal group, its determinant-one subgroup, the real form of signature $(p,r)$
$\operatorname{U}(V,h)$, $\operatorname{SU}(V,h)$, $U(p,r)$	the unitary group, its determinant-one subgroup, the real form of signature $(p,r)$
$\operatorname{Sp}(V,\omega)$, $Sp(2m,F)$	the symplectic group of a form, the standard split group
$Sp(n)$	the compact quaternionic unitary group, the compact real form of $Sp(2n,\mathbb{C})$
$GL(V)$, $SL(V)$	the general and special linear groups
$GO(V,q)$, $c$	the similarity group, its multiplier
$\tau_v$, $v^\perp$	the reflection in $v$, the hyperplane orthogonal to $v$
$\operatorname{diag}(I_p,-I_r,0_z)$	the normal form of a Hermitian form of signature $(p,r,z)$
$\operatorname{Pf}$	the Pfaffian
$A_n$, $B_n$, $C_n$, $D_n$	the four infinite families of simple Lie algebras