The Brauer–Wall Group and the Eightfold Way

Introduction

The classification of Clifford algebras has a group-theoretic form: the non-degenerate quadratic forms of a field, taken modulo hyperbolic planes, form the Witt group, and the assignment of its Clifford algebra to a form is additive. The target of this assignment is not the classical Brauer group but a graded refinement of it, the Brauer–Wall group, whose structure over the real numbers is cyclic of order eight. This article defines that group, computes it over $\mathbb{R}$ and $\mathbb{C}$, and states the Atiyah–Bott–Shapiro periodicity that identifies the theory of graded Clifford modules with real K-theory. The eight classes so obtained are the "eightfold way" of spinor types.

The Clifford classifications of the two preceding articles are used throughout: the graded tensor product $\mathrm{Cl}_{p,q}\hat\otimes\mathrm{Cl}_{r,s}\cong\mathrm{Cl}_{p+r,q+s}$, the eightfold table, and the identifications $\mathrm{Cl}_{8,0}\cong\mathrm{Cl}_{0,8}\cong M_{16}(\mathbb{R})$. Those results are not re-derived. The base is a field $F$ of characteristic not $2$ except where a statement is made over $\mathbb{R}$ or $\mathbb{C}$ explicitly.

The Witt-group material used here — quadratic spaces, orthogonal sums, hyperbolic planes, the Witt cancellation theorem — is standard and is being developed in parallel in the theory layer of this category. Only the shape of the results is needed: the Witt group $W(F)$ is the group of non-degenerate quadratic spaces modulo the hyperbolic ones under orthogonal sum, and it is generated by the one-dimensional forms with the relation that the hyperbolic plane is zero.

The Brauer Group

The ungraded prototype fixes the pattern that the graded group generalises.

Definition. A finite-dimensional $F$-algebra $A$ is central simple if its center is $F\cdot1$ and its only two-sided ideals are $\{0\}$ and $A$. Two central simple algebras are Morita equivalent if their categories of modules are equivalent; equivalently, if $A\cong M_m(D)$ and $B\cong M_n(D)$ for the same division algebra $D$. The Brauer group $\mathrm{Br}(F)$ is the set of Morita classes of central simple $F$-algebras, with product induced by the tensor product and unit the class of $F$.

Theorem. $\mathrm{Br}(F)$ is a group; if $F$ is algebraically closed then $\mathrm{Br}(F)=0$; and

$$ \mathrm{Br}(\mathbb{R})\cong\mathbb{Z}/2, $$

with the two classes represented by $\mathbb{R}$ and by $\mathbb{H}$. The class of $\mathbb{C}$ is not defined, because $\mathbb{C}$ is not central over $\mathbb{R}$.

Proof. Associativity and the unit are those of the tensor product; the inverse of the class of $A=M_n(D)$ is the class of the opposite algebra $A^{\mathrm{op}}$, since $A\otimes A^{\mathrm{op}}\cong M_{n^2}(F)$ is Morita trivial. The classification over $\mathbb{R}$ is the Frobenius–Wedderburn theorem: the finite-dimensional real division algebras are $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{H}$, and only $\mathbb{R}$ and $\mathbb{H}$ are central over $\mathbb{R}$.

The Brauer group is the right home for the ungraded Clifford algebra: an even-dimensional non-degenerate Clifford algebra is central simple, and its Morita class depends on the discriminant and the Hasse invariant of the form. But it discards the $\mathbb{Z}/2$-grading, and the grading is what the spin representations use. Refining the Brauer group by the grading is therefore necessary.

Graded Algebras and Graded Morita Equivalence

Definition. A graded algebra is an $F$-algebra $A$ together with a direct sum decomposition $A=A^0\oplus A^1$ such that $A^iA^j\subseteq A^{i+j}$ for $i,j\in\mathbb{Z}/2$. It is graded central simple if its graded center — the set of homogeneous elements $z$ with $zx=(-1)^{|x||z|}xz$ for every homogeneous $x$ — is exactly $F$, and it has no proper nonzero graded two-sided ideals. The graded tensor product $A\hat\otimes B$ has underlying space $A\otimes B$ with product

$$ (a\otimes b)(a'\otimes b')=(-1)^{|b||a'|}(aa')\otimes(bb'). $$

A Clifford algebra is the model example: $\mathrm{Cl}(V,q)$ is graded central simple precisely when $q$ is non-degenerate, and the graded tensor product of two non-degenerate Clifford algebras is the Clifford algebra of the orthogonal sum.

Definition. Two graded central simple algebras are graded Morita equivalent if their categories of graded modules are equivalent. The Brauer–Wall group $\mathrm{BW}(F)$ is the set of graded Morita classes of finite-dimensional graded central simple $F$-algebras, with product induced by the graded tensor product and unit the class of $F$ concentrated in degree $0$.

Proposition. $\mathrm{BW}(F)$ is an abelian group, and it contains a copy of $\mathrm{Br}(F)$, namely the subgroup of classes of the trivially graded central simple algebras. The classes of the non-degenerate Clifford algebras are elements of $\mathrm{BW}(F)$.

Proof. The graded tensor product is associative and commutative up to a canonical graded isomorphism — the sign in the product only affects the isomorphism, not the graded Morita class — and it has the unit $F$. Inverses exist by the graded analogue of the opposite-algebra argument. A trivially graded algebra is graded central simple exactly when it is central simple, and the graded tensor product of two trivially graded algebras is their ordinary tensor product, so the trivially graded classes form a subgroup isomorphic to $\mathrm{Br}(F)$. The Clifford class map is well defined on graded Morita classes because a non-degenerate Clifford algebra is graded central simple.

Remark. The difference between $\mathrm{Br}$ and $\mathrm{BW}$ is not merely bookkeeping. Neither $\mathbb{C}$ nor $\mathbb{D}$ is central simple over $\mathbb{R}$ — the center of $\mathbb{C}$ is $\mathbb{C}$ itself, and $\mathbb{D}=\mathbb{R}\times\mathbb{R}$ is a product of two fields — yet $\mathrm{Cl}_{0,1}=\mathbb{C}$ and $\mathrm{Cl}_{1,0}=\mathbb{D}$ are perfectly good graded central simple real algebras; the grading remembers the odd generator that the ungraded structure forgets. This is why the graded group is cyclic of order eight while the ungraded one is of order two.

The Clifford Classes

The non-degenerate quadratic forms give elements of $\mathrm{BW}(F)$ through their Clifford algebras.

Theorem. The assignment $q\mapsto[\mathrm{Cl}(V,q)]$ is a homomorphism

$$ W(F)\longrightarrow \mathrm{BW}(F) $$

from the Witt group of non-degenerate quadratic forms over $F$ to the Brauer–Wall group.

Proof. Orthogonal sums of forms give graded tensor products of their Clifford algebras, so the assignment is additive; the hyperbolic plane has the form $\operatorname{diag}(+1,-1)$ and Clifford algebra $\mathrm{Cl}_{1,1}\cong M_2(F)$, which is graded Morita trivial since it is a full matrix algebra and the module category is unchanged. By Witt cancellation, any two forms that represent the same element of $W(F)$ differ by hyperbolic planes, and therefore have graded Morita equivalent Clifford algebras.

So the classification of Clifford algebras is a specialisation of the classification of elements of $\mathrm{BW}(F)$, and the eightfold structure of the previous article is the structure of the subgroup generated by the one-dimensional forms.

Lemma. Let $q$ be a one-dimensional non-degenerate form over a field $F$ of characteristic not two, $q(x)=ax^{2}$ with $a\in F^{\times}$, and let $\mathrm{Cl}(q)=F[t]/(t^{2}-a)$ be its Clifford algebra. The graded Morita class $[\mathrm{Cl}(q)]\in\mathrm{BW}(F)$ depends only on the square class $aF^{\times2}$.

Proof. Two forms $ax^{2}$ and $bx^{2}$ with $b=ac^{2}$ are isometric by the linear substitution $x\mapsto cx$, hence have isomorphic Clifford algebras, and the graded Morita class depends only on the isomorphism class.

Remark. The assignment $\langle a\rangle\mapsto[\mathrm{Cl}(q_a)]$ is additive on the free abelian group generated by the square classes — the orthogonal sum $\langle a\rangle\perp\langle b\rangle$ has Clifford algebra the graded tensor product $\mathrm{Cl}(q_a)\hat\otimes\mathrm{Cl}(q_b)$, whose class is the sum — and it descends to the Witt group $W(F)$ because the relation $\langle a\rangle+\langle -a\rangle=0$ in $W(F)$ is matched by $[\mathrm{Cl}(q_a)]+[\mathrm{Cl}(q_{-a})]=0$ in $\mathrm{BW}(F)$; this is the content of the theorem above in the one-dimensional case. It is not a function of the product in $F^{\times}/F^{\times2}$: the one-dimensional forms generate $W(F)$ freely modulo the hyperbolic relations, and the product square class is a different quotient.

Example. The classes of the number systems in $\mathrm{BW}(\mathbb{R})\cong\mathbb{Z}/8$ are read from the class index $d\bmod8$: $\mathbb{R}$ has class $0$, $\mathbb{C}=\mathrm{Cl}_{0,1}$ class $7$, $\mathbb{D}=\mathrm{Cl}_{1,0}$ class $1$, $\mathbb{H}=\mathrm{Cl}_{0,2}$ class $6$, the four-dimensional split quaternions $\mathrm{Cl}_{1,1}\cong\mathrm{Cl}_{2,0}\cong M_2(\mathbb{R})$ class $0$ or $2$ according to the grading they carry, and $\mathbb{B}\cong\mathrm{Cl}_{3,0}$ class $3$. The two descriptions of $M_2(\mathbb{R})$ are not interchangeable here: the algebras are isomorphic but the gradings differ, and the class is an invariant of the graded algebra, so $\mathrm{Cl}_{1,1}$ lies in the trivial class while $\mathrm{Cl}_{2,0}$ does not. The classes add under the graded tensor product, so for instance $\mathbb{B}\hat\otimes\mathbb{B}\cong\mathrm{Cl}_{3,0}\hat\otimes\mathrm{Cl}_{3,0}\cong\mathrm{Cl}_{6,0}$ has class $6$, the class of $\mathbb{H}$, in agreement with $\mathrm{Cl}_{6,0}\cong M_4(\mathbb{H})$.

The Brauer–Wall Group of the Reals

Over $\mathbb{R}$ the group is cyclic of order eight, and the eightfold table is its explicit form.

Theorem (Wall). $\mathrm{BW}(\mathbb{R})\cong\mathbb{Z}/8$ and $\mathrm{BW}(\mathbb{C})\cong\mathbb{Z}/2$.

Proof. Over $\mathbb{R}$ the eight algebras $\mathrm{Cl}_{0,n}$ for $n=0,\dots,7$ are graded central simple and pairwise not graded Morita equivalent, by the eightfold table: their underlying algebras are

$$ \mathbb{R},\quad \mathbb{C},\quad \mathbb{H},\quad \mathbb{H}\times\mathbb{H},\quad M_2(\mathbb{H}),\quad M_4(\mathbb{C}),\quad M_8(\mathbb{R}),\quad M_8(\mathbb{R})\times M_8(\mathbb{R}), $$

and they are pairwise not graded Morita equivalent, since the graded Morita class of a graded central simple algebra is determined by its graded division algebra and the eight graded division algebras here are distinct. The graded tensor product gives

$$ \mathrm{Cl}_{0,m}\,\hat{\otimes}\,\mathrm{Cl}_{0,n}\cong \mathrm{Cl}_{0,m+n}, $$

so the classes of the $\mathrm{Cl}_{0,n}$ form a cyclic group generated by the class of $\mathrm{Cl}_{0,1}=\mathbb{C}$. Finally $\mathrm{Cl}_{0,8}=M_{16}(\mathbb{R})$ is graded Morita trivial, since a full matrix algebra with its standard grading has the same graded module category as $F$ up to the two parity shifts; hence the group is exactly $\mathbb{Z}/8$. Over $\mathbb{C}$ every non-degenerate form is hyperbolic or of dimension one, and $\mathrm{Cl}_1(\mathbb{C})\cong\mathbb{C}\times\mathbb{C}$ has square the trivial class, so the group is $\mathbb{Z}/2$.

Corollary. The nine algebras $\mathrm{Cl}_{0,0},\dots,\mathrm{Cl}_{0,8}$ represent the eight classes in order, with $\mathrm{Cl}_{0,8}$ returning to the unit class; and the mixed algebras $\mathrm{Cl}_{p,q}$ represent the same eight classes with the class index $p-q$ modulo eight. The assignment

$$ \mathbb{Z}/8\longrightarrow \mathrm{BW}(\mathbb{R}), \qquad d \longmapsto [\mathrm{Cl}_{p,q}],\quad p-q\equiv d, $$

is an isomorphism; and the trivially graded central simple real algebras form the two-element subgroup of $\mathbb{Z}/8$, the copy inside $\mathrm{BW}(\mathbb{R})$ of $\mathrm{Br}(\mathbb{R})=\mathbb{Z}/2$, whose two elements are the class of $\mathbb{R}$ and the class of $\mathbb{H}$ concentrated in degree $0$ (the class $4$, not the class $6$ carried by $\mathrm{Cl}_{0,2}$).

Remark. The existence of an element of order eight is a genuinely real phenomenon. Over an algebraically closed field the Brauer–Wall group has order two, and over a general field its order is governed by the arithmetic of the quadratic forms over that field; the group $\mathbb{Z}/8$ is an arithmetic invariant of $\mathbb{R}$, built from its two square classes together with the period-eight iteration of the sign in the Clifford pattern.

The Eightfold Way of Spinor Types

The order of $\mathrm{BW}(\mathbb{R})$ is the period of the kind of spinor representation, and it is conventional to display it as the sequence of division algebras.

Definition. Let $A$ be a graded central simple real algebra and let $M$ be its irreducible graded module. The type of $M$ is the division algebra $D$ with $A\cong M_k(D)$; it is $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$. Write $\varepsilon_n\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$ for the type of the irreducible $\mathrm{Cl}_{0,n}$-module.

Theorem (the eightfold sequence). The types are

$$ \mathbb{R},\ \mathbb{C},\ \mathbb{H},\ \mathbb{H},\ \mathbb{H},\ \mathbb{C},\ \mathbb{R},\ \mathbb{R}, $$

for $n=0,1,\dots,7$, and then repeat with period eight: $\varepsilon_{n+8}=\varepsilon_n$.

Proof. Read the division algebra column from the table of $\mathrm{Cl}_{0,n}$: $\mathrm{Cl}_{0,0}=\mathbb{R}$, $\mathrm{Cl}_{0,1}=\mathbb{C}$, $\mathrm{Cl}_{0,2}=\mathbb{H}$, $\mathrm{Cl}_{0,3}=\mathbb{H}\times\mathbb{H}$, $\mathrm{Cl}_{0,4}=M_2(\mathbb{H})$, $\mathrm{Cl}_{0,5}=M_4(\mathbb{C})$, $\mathrm{Cl}_{0,6}=M_8(\mathbb{R})$, $\mathrm{Cl}_{0,7}=M_8(\mathbb{R})\times M_8(\mathbb{R})$. The type of each is respectively $\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{H},\mathbb{H},\mathbb{C},\mathbb{R},\mathbb{R}$, and period eight is Bott periodicity, $\mathrm{Cl}_{0,n+8}\cong M_{16}(\mathrm{Cl}_{0,n})$, which does not change the division algebra.

The sequence is the precise form of the statement that spinors are real in dimensions $8k$, $8k+6$ and $8k+7$, complex in dimensions $8k+1$ and $8k+5$, and quaternionic in dimensions $8k+2,8k+3,8k+4$. The sequence for the family $\mathrm{Cl}_{n,0}$ is read from the same cyclic list in reverse, namely $\mathbb{R},\mathbb{R},\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{H},\mathbb{H},\mathbb{C}$ for $n=0,\dots,7$; equivalently the type of $\mathrm{Cl}_{n,0}$ equals the type of $\mathrm{Cl}_{0,8-n}$, so the two definite families are interchanged by the reversal $d\mapsto-d\bmod8$ of the class index, which is the effect of replacing the form by its negative. The two sequences correspond to the same eightfold periodicity read from the two definite signs.

Atiyah–Bott–Shapiro Periodicity

The correspondence between Clifford modules and K-theory makes the eightfold way an instance of topological periodicity.

Definition. Let $\mathrm{M}_n$ denote the Grothendieck group of finite-dimensional $\mathbb{Z}/2$-graded left $\mathrm{Cl}_{0,n}$-modules: the free abelian group on the isomorphism classes of graded modules modulo the additivity relations $[M\oplus N]=[M]+[N]$ and the relation $[\Pi M]=-[M]$, where $\Pi M$ is the same module with the parity of the grading reversed. The direct sum makes $\mathrm{M}_n$ an abelian group, and the parity shift acts on it by negation.

Theorem (Atiyah–Bott–Shapiro). For each $n$ there is a natural isomorphism

$$ \mathrm{M}_n\cong \mathrm{KO}^{-n}(\mathrm{pt}), $$

and both sides are periodic of period eight:

$n \bmod 8$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
$\mathrm{M}_n\cong\mathrm{KO}^{-n}(\mathrm{pt})$	$\mathbb{Z}$	$\mathbb{Z}/2$	$\mathbb{Z}/2$	$0$	$\mathbb{Z}$	$0$	$0$	$0$
type $\varepsilon_n$	$\mathbb{R}$	$\mathbb{C}$	$\mathbb{H}$	$\mathbb{H}$	$\mathbb{H}$	$\mathbb{C}$	$\mathbb{R}$	$\mathbb{R}$

Proof sketch. The graded module categories of the eight algebras $\mathrm{Cl}_{0,n}$ are read from the eightfold table: the simple graded modules are those of the matrix factors $M_k(D)$ with a compatible grading, the parity shift acts by $[\Pi M]=-[M]$, and carrying out the computation in each of the eight cases gives the row displayed. The coincidence with $\mathrm{KO}^{-n}(\mathrm{pt})$ is the Atiyah–Bott–Shapiro periodicity theorem, which identifies a graded Clifford module with the algebraic model of a real vector bundle with Clifford multiplication over a point.

Remark. The group $\mathrm{M}_n$ is the Grothendieck group of the graded module category of $\mathrm{Cl}_{0,n}$ with the parity relation $[\Pi M]=-[M]$, and its values are those of the row above: $\mathbb{Z}$ in dimensions $0$ and $4$, where one free generator survives, $\mathbb{Z}/2$ in dimensions $1$ and $2$, and zero in dimensions $3$, $5$, $6$ and $7$. The periodicity $\mathrm{M}_{n+8}\cong\mathrm{M}_n$ is Bott periodicity, and the vanishing and the torsion in the middle dimensions come from the extension relations of the graded module category together with the parity relation, read case by case from the eightfold table. The topological consequences — that the KO-groups of a point are of period eight, and that the index of a Clifford-linear elliptic operator is a KO-class — belong to the applications of the category.

Summary

The classical Brauer group $\mathrm{Br}(F)$ classifies central simple algebras up to Morita equivalence, and over $\mathbb{R}$ it is $\mathbb{Z}/2$. Refining by the $\mathbb{Z}/2$-grading gives the Brauer–Wall group $\mathrm{BW}(F)$, which classifies graded central simple algebras up to graded Morita equivalence under the graded tensor product. The Clifford algebra of a non-degenerate quadratic form is graded central simple, and the assignment $q\mapsto\mathrm{Cl}(V,q)$ is a homomorphism $W(F)\to\mathrm{BW}(F)$; the hyperbolic plane maps to a matrix algebra and hence to the trivial class.

Over $\mathbb{R}$ the Brauer–Wall group is cyclic of order eight, $\mathrm{BW}(\mathbb{R})\cong\mathbb{Z}/8$, generated by $\mathrm{Cl}_{0,1}=\mathbb{C}$, with $\mathrm{Cl}_{0,8}\cong M_{16}(\mathbb{R})$ returning to the unit; over $\mathbb{C}$ it is $\mathbb{Z}/2$. The eight classes are represented by $\mathrm{Cl}_{0,n}$ for $n=0,\dots,7$, and the class index is $d=p-q$ modulo eight. The types of the irreducible modules form the eightfold sequence $\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{H},\mathbb{H},\mathbb{C},\mathbb{R},\mathbb{R}$, and the Grothendieck group of graded Clifford modules is the eightfold periodic group $\mathrm{KO}^{-n}(\mathrm{pt})$ of Atiyah–Bott–Shapiro.

Summary of Notation

Symbol	Meaning
$F$	Field of characteristic not $2$
$\mathrm{Br}(F)$	Brauer group of central simple $F$-algebras up to Morita equivalence; $\mathrm{Br}(\mathbb{R})=\mathbb{Z}/2$
$A^{\mathrm{op}}$	Opposite algebra, inverse of the class of $A$ in $\mathrm{Br}(F)$
$A=A^0\oplus A^1$	Grading of a graded algebra
$\hat\otimes$	Graded tensor product, $(a\otimes b)(a'\otimes b')=(-1)^{\|b\|\|a'\|}(aa'\otimes bb')$
$\mathrm{BW}(F)$	Brauer–Wall group of graded central simple algebras up to graded Morita equivalence
$[\mathrm{Cl}(V,q)]$	Class of a Clifford algebra in $\mathrm{BW}(F)$
$W(F)$	Witt group of non-degenerate quadratic forms
$q\mapsto[\mathrm{Cl}(V,q)]$	Homomorphism $W(F)\to\mathrm{BW}(F)$
$\mathrm{Cl}_{0,n}$	Negative definite Clifford algebra, representatives of the eight classes over $\mathbb{R}$
$\mathrm{BW}(\mathbb{R})\cong\mathbb{Z}/8$, $\mathrm{BW}(\mathbb{C})\cong\mathbb{Z}/2$	Wall's computation
$\varepsilon_n$	Type of the irreducible $\mathrm{Cl}_{0,n}$-module: $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$
$\mathrm{M}_n$	Grothendieck group of graded $\mathrm{Cl}_{0,n}$-modules
$\mathrm{KO}^{-n}(\mathrm{pt})$	Real K-theory of a point, periodic of period eight