List of Clifford Algebras and Spin Groups

Introduction

This article lists the Clifford algebras and the spin groups of the corpus: the algebras $\mathrm{Cl}(V,q)$ and their real signature forms $\mathrm{Cl}_{p,q}$, the periodicity that classifies them, the Clifford, Pin and Spin groups inside their unit groups, and the spinor modules on which they act. The property that a Clifford algebra has is that it is generated by a quadratic space subject to the single relation $v^2 = q(v)\cdot 1$; the property that a spin group has is that it lies inside the units of the Clifford algebra and acts on the quadratic space by isometries, giving a double cover of the orthogonal group.

Every entry points to the article that introduces the algebra, the group or the module. The article introduces nothing and proves nothing: it records the relation, the periodicity class, the cover and the module type as the introducing articles give them, and it neither restates a definition nor gives a proof.

The article records examples and non-examples side by side. Beside the algebras that are simple or central simple it lists the cases in which the structure degenerates: $\mathrm{Cl}_{1,0} \cong F \times F$ is a product of two algebras and has zero divisors, so a Clifford algebra is not a division algebra in general; the Clifford algebra of a degenerate form has a nilpotent radical and is not semisimple; the Pin group covers the disconnected group $O(V,q)$, so it does not inherit connectedness; the spin representation is a representation of $\mathrm{Spin}(n)$ and not of $SO(n)$; and a spin structure on a manifold is a bundle-theoretic datum and not a Clifford algebra, each with the failure named and the article that records it.

The Clifford Algebra and Its Structure

The Clifford algebra is the universal solution of the relation $v^2 = q(v)$, and its parity grading and graded tensor product are the operations on which the whole layer rests.

Object or property	The statement	Introduced in
the Clifford algebra $\mathrm{Cl}(V,q)$	the quotient of the tensor algebra by the ideal generated by $v \otimes v - q(v)\cdot 1$; the universal algebra of the relation	Clifford Algebras
the universal property	every linear map to an algebra with the relation factors uniquely through $\mathrm{Cl}(V,q)$	Clifford Algebras
the fundamental relation	$uv + vu = 2B(u,v)\cdot 1$; the polarised form of the defining relation	Clifford Algebras; Quadratic Forms and Polarisation
the parity grading $\mathrm{Cl} = \mathrm{Cl}^0 \oplus \mathrm{Cl}^1$	the even and odd parts; $\mathrm{Cl}^0$ is a subalgebra, the even Clifford algebra	Clifford Algebras; Clifford Algebras in Finite Dimensions
the grade involution $\alpha$	$v \mapsto -v$, with $\alpha(x) = (-1)^kx$ on the degree-$k$ part	Clifford Algebras; The Clifford, Pin and Spin Groups with Signed Inner Conjugation
reversion $x \mapsto x^r$ and Clifford conjugation $x^{\natural} = \alpha(x^r)$	the two anti-involutions; reversion fixes the vectors, conjugation negates them	The Clifford, Pin and Spin Groups with Signed Inner Conjugation; Clifford Algebras in Finite Dimensions
the graded tensor product	$\mathrm{Cl}(V_1,q_1)\,\hat\otimes\,\mathrm{Cl}(V_2,q_2) \cong \mathrm{Cl}(V_1 \oplus V_2, q_1 \perp q_2)$	Clifford Algebras; Bott Periodicity and the Classification
the volume element $\omega = e_1\cdots e_n$	$\omega^2 = (-1)^{n(n-1)/2}\prod_i q(e_i)$; it decides the centre in odd dimension	Clifford Algebras in Finite Dimensions; Bott Periodicity and the Classification
the exterior algebra as the case $q = 0$	$\Lambda(V) = \mathrm{Cl}(V,0)$; the Clifford algebra is a deformation of it	Clifford Algebras
the degenerate case	a degenerate $q$ gives a Clifford algebra whose radical is an ideal; the non-degenerate factor and the nilpotent factor are separated	Clifford Algebras in Finite Dimensions; Degenerate Clifford Algebras and the Radical
the bivectors and the Lie algebra	$\mathrm{SO}(V,q) \cong \Lambda^2V$ inside the Clifford algebra; the bivectors form a Lie algebra under the commutator	The Clifford Algebra as a Lie Algebra
the trace and the centre	the centre is $F$, $F(\sqrt\delta)$ or $F \times F$ according to the parity of $n$ and the class of $\delta = \omega^2$	Bott Periodicity and the Classification
the spin factor	the Jordan algebra $JSpin(V) = R\cdot1 \oplus V$; its Clifford envelope	Spin Factors and the Clifford Envelope with Inner Conjugation

The Classification and the Periodicity

The complex classification is periodic of period two and the real one of period eight; the rank-eight algebra is a matrix algebra over the base field, and from it the whole table follows.

Statement	The content	Introduced in
complex periodicity	$\mathbb{C}\mathrm{l}_{2m} \cong M_{2^m}(\mathbb{C})$ and $\mathbb{C}\mathrm{l}_{2m+1} \cong M_{2^m}(\mathbb{C}) \times M_{2^m}(\mathbb{C})$; period two	Bott Periodicity and the Classification; Spin Representations and Clifford Modules with Inner Conjugation
real periodicity	the sequence $\mathrm{Cl}_{p,q}$ is periodic of period eight in $p - q$	Bott Periodicity and the Classification
the periodicity algebra	$\mathrm{Cl}_{0,8} \cong \mathrm{Cl}_{8,0} \cong M_{16}(F)$	Bott Periodicity and the Classification
the one-, two- and three-dimensional cases	$\mathrm{Cl}_{0,1} \cong \mathbb{C}$ and $\mathrm{Cl}_{1,0} \cong \mathbb{D} = F\times F$; $\mathrm{Cl}_{0,2} \cong \mathbb{H}$; $\mathrm{Cl}_{3,0} \cong \mathbb{B} = M_2(\mathbb{C})$	The Low-Dimensional Classification; The Number Systems as Clifford Algebras
the eightfold class of a form	the division algebra $D$ of the irreducible module is read from $d = p - q \bmod 8$	Bott Periodicity and the Classification
the module type over $\mathbb{R}$	the irreducible module is real for $d \equiv 0,1,2$; complex for $d \equiv 3,7$; quaternionic for $d \equiv 4,5,6$ (mod $8$)	Real Spinors and Reality Conditions with Inner Conjugation; Bott Periodicity and the Classification
the dimension of the spinor module	$\dim_\mathbb{R}\Delta = k\dim_\mathbb{R}D$ from $\mathrm{Cl}_{p,q} \cong M_k(D)$, and $2^{\lfloor n/2\rfloor}$ over $\mathbb{C}$	Bott Periodicity and the Classification; Spin Representations and Clifford Modules with Inner Conjugation
the definite family $\mathrm{Cl}_{n,0}$	the type is read from $n \bmod 8$; in $\mathrm{Cl}_{0,n}$ from $-n \equiv 8 - n \bmod 8$	Real Spinors and Reality Conditions with Inner Conjugation

The eightfold table of the reality conditions of the spinor module, as recorded in the corpus.

$d \bmod 8$	type of $\Delta$	reality structure	spinors
$0$	real	$J^2 = +1$	Majorana
$1$	real	$J^2 = +1$	Majorana
$2$	real	$J^2 = +1$	Majorana
$3$	complex	none	Weyl, conjugate pair
$4$	quaternionic	$J^2 = -1$	symplectic
$5$	quaternionic	$J^2 = -1$	symplectic
$6$	quaternionic	$J^2 = -1$	symplectic
$7$	complex	none	conjugate pair

The Clifford, Pin and Spin Groups with Signed Inner Conjugation

Inside the units of the Clifford algebra sit the Clifford group, the Pin group and the Spin group; the signed inner conjugation action sends them onto the orthogonal groups, with kernel $\{\pm1\}$.

Group or statement	The property it has	Introduced in
the Clifford group $\Gamma(V,q)$	the units preserving the space of vectors under the signed inner conjugation action $\mathrm{Ad}^{\alpha}_x(v) = \alpha(x)vx^{-1}$	The Clifford, Pin and Spin Groups with Signed Inner Conjugation
the Clifford norm $N(x) = x x^{\natural}$	the norm on the Clifford group; the Pin group is the part with $N = \pm1$	The Clifford, Pin and Spin Groups with Signed Inner Conjugation
the Pin group $\mathrm{Pin}(V,q)$	$\{x \in \Gamma : N(x) = \pm1\}$; it acts on the space by reflections and double covers $O(V,q)$ over $\mathbb{R}$	The Clifford, Pin and Spin Groups with Signed Inner Conjugation
the Spin group $\mathrm{Spin}(V,q)$	$\mathrm{Pin} \cap \mathrm{Cl}^0$; the double cover of $SO(V,q)$	The Clifford, Pin and Spin Groups with Signed Inner Conjugation
the reflection $\rho_u$	$\rho_u(v) = v - 2g(v,u)q(u)^{-1}u$; the signed inner conjugation action of a vector	The Clifford, Pin and Spin Groups with Signed Inner Conjugation
Cartan–Dieudonné and the reflection length	the Pin group is generated by the vectors of norm $\pm1$; the number of reflections is the parity that separates Pin from Spin	The Clifford, Pin and Spin Groups with Signed Inner Conjugation; Isometries and Orthogonal Transformations
the signature cases $\mathrm{Pin}(p,q)$, $\mathrm{Spin}(p,q)$	the groups of the form $q = \operatorname{diag}(+1^p,-1^q)$; the definite case is written $\mathrm{Pin}(n)$, $\mathrm{Spin}(n)$	The Clifford, Pin and Spin Groups with Signed Inner Conjugation
the low-dimensional spin groups	$\mathrm{Spin}(2) \cong U(1)$, $\mathrm{Spin}(3) \cong Sp(1) = SU(2)$, $\mathrm{Spin}(4) \cong Sp(1) \times Sp(1)$, $\mathrm{Spin}(5) \cong Sp(2)$, $\mathrm{Spin}(6) \cong SU(4)$	The Clifford, Pin and Spin Groups with Signed Inner Conjugation
the vector representation	$\rho(s)(v) = svs^{-1}$ on the vector space; the double cover of $SO(V,q)$	Spin Factors and the Clifford Envelope with Inner Conjugation; The Clifford, Pin and Spin Groups with Signed Inner Conjugation
the complex case	$\operatorname{Spin}(6) \cong SL(4,\mathbb{C})$ and $PO(6) \cong PGL(4)$; the split real form $\operatorname{Spin}(3,3) \cong SL(4,\mathbb{R})$	Projective Geometry; The Clifford, Pin and Spin Groups with Signed Inner Conjugation

The Spinor Modules

A Clifford module is a module over the Clifford algebra, and the spinor module is the irreducible one, built explicitly from a maximal isotropic subspace; its parity splitting is the chirality that defines the Weyl spinors.

Object or property	The statement	Introduced in
a Clifford module	a left module over $\mathrm{Cl}(V,q)$, with the $\mathbb{Z}/2$-grading it inherits	Spin Representations and Clifford Modules with Inner Conjugation
the spinor module from a maximal isotropic subspace	$\Delta = \Lambda^\bullet W$ for $V = W \oplus W'$; the Clifford action $c(w + w') = \varepsilon(w) + \iota(w')$	Spin Representations and Clifford Modules with Inner Conjugation
the dimension of $\Delta$	$2^m$ over $\mathbb{C}$ for $n = 2m$ or $2m+1$; the module is $D^k$ in the real cases	Spin Representations and Clifford Modules with Inner Conjugation; Bott Periodicity and the Classification
chirality and the half-spin representations	in even dimension $\Delta = \Delta_+ \oplus \Delta_-$ by the volume element; the half-spin representations $\rho_\pm$	Spin Representations and Clifford Modules with Inner Conjugation
the infinitesimal spin representation	$d\rho(v \wedge w) = \tfrac14[c(v),c(w)]$ on $\mathrm{SO}(V,q) \cong \Lambda^2V$	Spin Representations and Clifford Modules with Inner Conjugation
the real, complex and quaternionic spinors	the reality structures by $d \bmod 8$; Majorana, Weyl and Majorana–Weyl spinors	Real Spinors and Reality Conditions with Inner Conjugation
the biquaternion spinor module	the spinor module of the biquaternion algebra $\mathbb{B} \cong M_2(\mathbb{C})$; its defining module is of complex type	Biquaternion Spin Geometry; The Clifford Structure of the Biquaternion Algebra
the spinor bundle and the Dirac operator	the geometric realisation of a Clifford module over a manifold	Spin Geometry; Dirac Differential Operators

Non-examples and Warnings

Object	Why the expected statement fails	Introduced in
the Clifford algebra $\mathrm{Cl}_{1,0} \cong F \times F$	it is a product of two algebras and has zero divisors; a Clifford algebra is not a division algebra in general	The Low-Dimensional Classification; List of Division Algebras
the Clifford algebra of a degenerate form	its radical is a non-trivial nilpotent ideal and the algebra is not semisimple	Degenerate Clifford Algebras and the Radical; Clifford Algebras in Finite Dimensions
the Pin group	it double covers $O(V,q)$, which is disconnected; the connected component is covered by Spin	The Clifford, Pin and Spin Groups with Signed Inner Conjugation
the spin representation of $SO(n)$	it is a representation of $\mathrm{Spin}(n)$ and not of $SO(n)$: $\rho(-1) = -\mathrm{id}_S$, so the sign of the double cover survives; a half-spin representation can have a larger kernel, as $\mathrm{Spin}(4) \cong Sp(1)\times Sp(1)$ on $\Delta_+$ with kernel $\{1\}\times Sp(1)$	Spin Representations and Clifford Modules with Inner Conjugation
the identification $\mathrm{Spin}(n) \cong SO(n)$	it fails for every $n \geq 3$; the covering is two-to-one and non-trivial	The Clifford, Pin and Spin Groups with Signed Inner Conjugation; Matrix Groups and Classical Groups
the biquaternion algebra as a division algebra	$\mathbb{B} \cong M_2(\mathbb{C})$ is not a division algebra; it is a Clifford algebra of complex type	The Clifford Structure of the Biquaternion Algebra; List of Division Algebras
the odd-dimensional Clifford algebra	for $n$ odd and $\delta$ a square it is a product $A \times A$ with two distinct irreducible modules, so the spinor module is not unique; the volume element exchanges the two	Bott Periodicity and the Classification
a spin structure on a manifold	it is a bundle-theoretic datum, a principal $\mathrm{Spin}(n)$-bundle lifting the frame bundle, and not a Clifford algebra	Spin Geometry

Objects that a reader may expect in a list of Clifford algebras and spin groups, and does not find here.

Object	Why it is not listed	Introduced in
the orthogonal group $O(V,q)$	it is the group that Pin covers and is catalogued with the classical geometric groups	List of Classical Geometric Groups
the projective orthogonal group $PO(V,q)$	it is the quotient by the centre and is catalogued with the projective groups	List of Projective Geometric Groups
the octonions and the exceptional groups	the octonion algebra is not a Clifford algebra of the standard kind and is recorded with the norms and the exceptional Lie groups; $G_2 = \operatorname{Aut}(\mathbb{O})$	Quadratic Forms over Algebras and Norms; Octonions and the Exceptional Lie Groups
the Dirac and Weyl operators	they are differential operators on a spin manifold, catalogued with the differential operators	Dirac Differential Operators; Spin Geometry
the infinite-dimensional Clifford algebras and CAR	they carry a topology and a $C^*$-algebraic structure, outside the finite-dimensional classification	Infinite-Dimensional Clifford Algebras and CAR with Inner Conjugation

Summary

This article has listed the Clifford algebras and spin groups of the corpus: the algebra $\mathrm{Cl}(V,q)$ with its universal property, parity grading, grade involution, the two anti-involutions and the graded tensor product, the volume element and the bivectors that form $\mathrm{SO}(V,q)$; the classification by complex period two and real period eight, with the rank-eight algebra $M_{16}(F)$ and the eightfold table of real, complex and quaternionic spinors; the Clifford, Pin and Spin groups with the signed inner conjugation action, the double covers of the orthogonal groups and the low-dimensional identifications $\mathrm{Spin}(3) \cong Sp(1)$, $\mathrm{Spin}(4) \cong Sp(1) \times Sp(1)$, $\mathrm{Spin}(5) \cong Sp(2)$, $\mathrm{Spin}(6) \cong SU(4)$; and the spinor modules with their chirality splitting and reality conditions. Beside the examples stand the non-examples: the split algebra $F\times F$ with its zero divisors, the degenerate case with its nilpotent radical, the disconnected Pin group, the spin representation that belongs to $\mathrm{Spin}$ and not to $SO$, and the odd-dimensional algebra with two distinct irreducible modules. The list introduces and proves nothing; it is the index of the Clifford layer 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
$q$, $B$, $g$	a quadratic form, its polar form, the polar form written $g$
$\mathrm{Cl}(V,q)$	the Clifford algebra of $q$ on $V$
$v^2 = q(v)\cdot1$, $uv + vu = 2B(u,v)$	the defining and fundamental relations
$\mathrm{Cl}^0$, $\mathrm{Cl}^1$	the even and odd parts of the parity grading
$\alpha$, $x^r$, $x^{\natural}$	the grade involution, reversion, Clifford conjugation
$\hat\otimes$	the graded tensor product
$\mathrm{Cl}_{p,q}$	the Clifford algebra with $p$ positive and $q$ negative squares
$d = p - q$, $n = p + q$	the signature difference and the total dimension
$\omega = e_1\cdots e_n$, $\delta = \omega^2$	the volume element and its square
$\Gamma(V,q)$, $N(x) = x x^{\natural}$	the Clifford group and its norm
$\mathrm{Pin}(V,q)$, $\mathrm{Spin}(V,q)$	the Pin and Spin groups
$\rho_u$, $\mathrm{Ad}^{\alpha}_x$	the reflection and the signed inner conjugation action
$\Delta = \Lambda^\bullet W$, $\Delta_\pm$	the spinor module and its chiral halves
$c(v)$, $\rho$, $d\rho$	the Clifford action, the spin representation, its differential
$J$, $J^2 = \pm1$	the reality structure; real and quaternionic type
$\mathbb{D}$, $\mathbb{H}$, $\mathbb{B}$, $M_k(D)$	split complex, quaternion, biquaternion algebras; matrix algebras