My last article on the ten-fold way was a piece of research in progress — it only reached a nice final form in the comments. Since that made it rather hard to follow, let me try to present a more detailed and self-contained treatment here!

But if you’re in a hurry, you can click on this:

and get my poster for next week’s scientific advisory board meeting at the Centre for Quantum Technologies, in Singapore. That’s where I work in the summer, and this poster is supposed to be a terse introduction to the ten-fold way.

First we’ll introduce the ‘Brauer monoid’ of a field. This is a way of assembling all simple algebras over that field into a monoid: a set with an associative product and unit. One reason for doing this is that in quantum physics, physical systems are described by vector spaces that are representations of certain ‘algebras of observables’, which are sometimes simple (in the technical sense). Combining physical systems involves taking the tensor product of their vector spaces and also these simple algebras. This gives the multiplication in the Brauer monoid.

We then turn to a larger structure called the ‘super Brauer monoid’ or ‘Brauer–Wall monoid’. This is the ‘super’ or $<semantics>{\mathbb{Z}}_{2}<annotation\; encoding="application/x-tex">\backslash mathbb\{Z\}\_2</annotation></semantics>$-graded version of the same idea, which shows up naturally in physical systems containing both bosons and fermions. For the field $<semantics>\mathbb{R}<annotation\; encoding="application/x-tex">\backslash mathbb\{R\}</annotation></semantics>$, the super Brauer monoid has 10 elements. This gives a nice encapsulation of the ‘ten-fold way’ introduced in work on condensed matter physics. At the end I’ll talk about this particular example in more detail.

Actually elements of the Brauer monoid of a field are *equivalence classes* of simple algebras over this field. Thus, I’ll start by reminding you about simple algebras and the notion of equivalence we need, called ‘Morita equivalence’. Briefly, two algebras are Morita equivalent if they have the same category of representations. Since in quantum physics it’s the representations of an algebra that matter, this is sometimes the right concept of equivalence, even though it’s coarser than isomorphism.

## Review of algebra

We begin with some material that algebraists consider well-known.

### Simple algebras and division algebras

Let $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ be a field.

By an **algebra** over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ we will always mean a finite-dimensional associative unital $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$-algebra: that is, a finite-dimensional vector space $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ with an associative bilinear multiplication and a multiplicative unit $<semantics>1\in A<annotation\; encoding="application/x-tex">1\; \backslash in\; A</annotation></semantics>$.

An algebra $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ is **simple** if its only two-sided ideals are $<semantics>\{0\}<annotation\; encoding="application/x-tex">\backslash \{0\backslash \}</annotation></semantics>$ and $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ itself.

A **division algebra** over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ is an algebra $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ such that if $<semantics>a\ne 0<annotation\; encoding="application/x-tex">a\; \backslash ne\; 0</annotation></semantics>$ there exists $<semantics>b\in A<annotation\; encoding="application/x-tex">b\; \backslash in\; A</annotation></semantics>$ such that $<semantics>ab=ba=1<annotation\; encoding="application/x-tex">a\; b=\; b\; a\; =\; 1</annotation></semantics>$. Using finite-dimensionality the following condition is equivalent: if $<semantics>a,b\in A<annotation\; encoding="application/x-tex">a,b\; \backslash in\; A</annotation></semantics>$ and $<semantics>ab=0<annotation\; encoding="application/x-tex">a\; b\; =\; 0</annotation></semantics>$ then either $<semantics>a=0<annotation\; encoding="application/x-tex">a\; =\; 0</annotation></semantics>$ or $<semantics>b=0<annotation\; encoding="application/x-tex">b\; =\; 0</annotation></semantics>$.

A division algebra is automatically simple. More interestingly, by a theorem of Wedderburn, every simple algebra $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ is an algebra of $<semantics>n\times n<annotation\; encoding="application/x-tex">n\; \backslash times\; n</annotation></semantics>$ matrices with entries in some division algebra $<semantics>D<annotation\; encoding="application/x-tex">D</annotation></semantics>$ over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$. We write this as

$$<semantics>A\cong D[n]<annotation\; encoding="application/x-tex">\; A\; \backslash cong\; D[n]\; </annotation></semantics>$$

where $<semantics>D[n]<annotation\; encoding="application/x-tex">D[n]</annotation></semantics>$ is our shorthand for the algebra of $<semantics>n\times n<annotation\; encoding="application/x-tex">n\; \backslash times\; n</annotation></semantics>$ matrices with entries in $<semantics>D<annotation\; encoding="application/x-tex">D</annotation></semantics>$.

The center of an algebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ always includes a copy of $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$, the scalar multiples of $<semantics>1\in A<annotation\; encoding="application/x-tex">1\; \backslash in\; A</annotation></semantics>$. If $<semantics>D<annotation\; encoding="application/x-tex">D</annotation></semantics>$ is a division algebra, its center $<semantics>Z(D)<annotation\; encoding="application/x-tex">Z(D)</annotation></semantics>$ is a commutative algebra that’s a division algebra in its own right. So $<semantics>Z(D)<annotation\; encoding="application/x-tex">Z(D)</annotation></semantics>$ is field, and it’s a **finite extension** of $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$, meaning it contains $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ as a subfield and is a finite-dimensional algebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$.

If $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ is a simple algebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$, its center is isomorphic to the center of some $<semantics>D[n]<annotation\; encoding="application/x-tex">D[n]</annotation></semantics>$, which is just the center of $<semantics>D<annotation\; encoding="application/x-tex">D</annotation></semantics>$. So, the center of $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ is a field that’s a finite extension of $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$. We’ll need this fact when defining in the multiplication in the Brauer monoid.

**Example.** I’m mainly interested in the case $<semantics>k=\mathbb{R}<annotation\; encoding="application/x-tex">k\; =\; \backslash mathbb\{R\}</annotation></semantics>$. A theorem of Frobenius says the only division algebras over $<semantics>\mathbb{R}<annotation\; encoding="application/x-tex">\backslash mathbb\{R\}</annotation></semantics>$ are $<semantics>\mathbb{R}<annotation\; encoding="application/x-tex">\backslash mathbb\{R\}</annotation></semantics>$ itself, the complex numbers $<semantics>\u2102<annotation\; encoding="application/x-tex">\backslash mathbb\{C\}</annotation></semantics>$ and the quaternions $<semantics>\mathbb{H}<annotation\; encoding="application/x-tex">\backslash mathbb\{H\}</annotation></semantics>$. Of these, the first two are fields, while the third is noncommutative. So, the simple algebras over $<semantics>\mathbb{R}<annotation\; encoding="application/x-tex">\backslash mathbb\{R\}</annotation></semantics>$ are the matrix algebras $<semantics>\mathbb{R}[n]<annotation\; encoding="application/x-tex">\backslash mathbb\{R\}[n]</annotation></semantics>$, $<semantics>\u2102[n]<annotation\; encoding="application/x-tex">\backslash mathbb\{C\}[n]</annotation></semantics>$ and $<semantics>\mathbb{H}[n]<annotation\; encoding="application/x-tex">\backslash mathbb\{H\}[n]</annotation></semantics>$. The center of $<semantics>\mathbb{R}[n]<annotation\; encoding="application/x-tex">\backslash mathbb\{R\}[n]</annotation></semantics>$ and $<semantics>\mathbb{H}[n]<annotation\; encoding="application/x-tex">\backslash mathbb\{H\}[n]</annotation></semantics>$ is $<semantics>\mathbb{R}<annotation\; encoding="application/x-tex">\backslash mathbb\{R\}</annotation></semantics>$, while the center of $<semantics>\u2102[n]<annotation\; encoding="application/x-tex">\backslash mathbb\{C\}[n]</annotation></semantics>$ is $<semantics>\u2102<annotation\; encoding="application/x-tex">\backslash mathbb\{C\}</annotation></semantics>$, the only nontrivial finite extension of $<semantics>\mathbb{R}<annotation\; encoding="application/x-tex">\backslash mathbb\{R\}</annotation></semantics>$.

**Example.** The case $<semantics>k=\u2102<annotation\; encoding="application/x-tex">k\; =\; \backslash mathbb\{C\}</annotation></semantics>$ is more boring, because $<semantics>\u2102<annotation\; encoding="application/x-tex">\backslash mathbb\{C\}</annotation></semantics>$ is algebraically closed. Any division algebra $<semantics>D<annotation\; encoding="application/x-tex">D</annotation></semantics>$ over an algebraically closed field $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ must be $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ itself. (To see this, consider $<semantics>x\in D<annotation\; encoding="application/x-tex">x\; \backslash in\; D</annotation></semantics>$ and look at the smallest subring of $<semantics>D<annotation\; encoding="application/x-tex">D</annotation></semantics>$ containing $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ and $<semantics>x<annotation\; encoding="application/x-tex">x</annotation></semantics>$ and closed under taking inverses. This is a finite hence algebraic extension of $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$, so it must be $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$.) So if $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ is algebraically closed, the only simple algebras over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ are the matrix algebras $<semantics>k[n]<annotation\; encoding="application/x-tex">k[n]</annotation></semantics>$.

**Example.** The case where $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ is a finite field has a very different flavor. A theorem of Wedderburn and Dickson implies that any division algebra over a finite field $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ is a field, indeed a finite extension of $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$. So, the only simple algebras over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ are the matrix algebras $<semantics>F[n]<annotation\; encoding="application/x-tex">F[n]</annotation></semantics>$ where $<semantics>F<annotation\; encoding="application/x-tex">F</annotation></semantics>$ is a finite extension of $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$. Moreover, we can completely understand these finite extensions, since the finite fields are all of the form $<semantics>{\mathbb{F}}_{{p}^{n}}<annotation\; encoding="application/x-tex">\backslash mathbb\{F\}\_\{p^n\}</annotation></semantics>$ where $<semantics>p<annotation\; encoding="application/x-tex">p</annotation></semantics>$ is a prime and $<semantics>n=1,2,3,\dots <annotation\; encoding="application/x-tex">n\; =\; 1,2,3,\backslash dots</annotation></semantics>$, and the only finite extensions of $<semantics>{\mathbb{F}}_{{p}^{n}}<annotation\; encoding="application/x-tex">\backslash mathbb\{F\}\_\{p^n\}</annotation></semantics>$ are the fields $<semantics>{\mathbb{F}}_{{p}^{m}}<annotation\; encoding="application/x-tex">\backslash mathbb\{F\}\_\{p^m\}</annotation></semantics>$ where $<semantics>n<annotation\; encoding="application/x-tex">n</annotation></semantics>$ divides $<semantics>m<annotation\; encoding="application/x-tex">m</annotation></semantics>$.

### Morita equivalence and the Brauer group

Given an algebra $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ we define **$<semantics>\mathrm{Rep}(A)<annotation\; encoding="application/x-tex">Rep(A)</annotation></semantics>$** to be the category of left $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$-modules. We say two algebras $<semantics>A,B<annotation\; encoding="application/x-tex">A,\; B</annotation></semantics>$ over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ are **Morita equivalent** if $<semantics>\mathrm{Rep}(A)\simeq \mathrm{Rep}(B)<annotation\; encoding="application/x-tex">Rep(A)\; \backslash simeq\; Rep(B)</annotation></semantics>$. In this situation we write **$<semantics>A\simeq B<annotation\; encoding="application/x-tex">A\; \backslash simeq\; B</annotation></semantics>$**.

Isomorphic algebras are Morita equivalent, but this equivalence relation is more general; for example we always have $<semantics>A[n]\simeq A<annotation\; encoding="application/x-tex">A[n]\; \backslash simeq\; A</annotation></semantics>$, where $<semantics>A[n]<annotation\; encoding="application/x-tex">A[n]</annotation></semantics>$ is the algebra of $<semantics>n\times n<annotation\; encoding="application/x-tex">n\; \backslash times\; n</annotation></semantics>$ matrices with entries in $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$.

We’ve seen that if $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ is simple, $<semantics>A\cong D[n]<annotation\; encoding="application/x-tex">A\; \backslash cong\; D[n]</annotation></semantics>$, and this implies
$<semantics>A\simeq D[n]<annotation\; encoding="application/x-tex">A\; \backslash simeq\; D[n]</annotation></semantics>$. On the other hand, we have $<semantics>D[n]\simeq D<annotation\; encoding="application/x-tex">D[n]\; \backslash simeq\; D</annotation></semantics>$. So, every simple algebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ is Morita equivalent to a division algebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$.

As a set, the Brauer monoid of $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ will simply be the set of Morita equivalence classes of simple algebras over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$. By what I just said, this is also the set of Morita equivalence classes of division algebras over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$. The trick will be defining multiplication in the Brauer monoid. For this we need to think about tensor products of algebras.

The tensor product of two algebras $<semantics>A,B<annotation\; encoding="application/x-tex">A,B</annotation></semantics>$ over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ is another algebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$, which we’ll write as $<semantics>A{\otimes}_{k}B<annotation\; encoding="application/x-tex">A\; \backslash otimes\_k\; B</annotation></semantics>$. This gets along with Morita equivalence:

$$<semantics>A\simeq A\prime \phantom{\rule{thickmathspace}{0ex}}\mathrm{and}\phantom{\rule{thickmathspace}{0ex}}B\simeq B\prime \phantom{\rule{thickmathspace}{0ex}}\Rightarrow \phantom{\rule{thickmathspace}{0ex}}A{\otimes}_{k}A\prime \simeq B{\otimes}_{k}B\prime <annotation\; encoding="application/x-tex">\; A\; \backslash simeq\; A\text{\'}\; \backslash ;\; and\; \backslash ;\; B\; \backslash simeq\; B\text{\'}\; \backslash ;\; \backslash implies\; \backslash ;\; A\; \backslash otimes\_k\; A\text{\'}\; \backslash simeq\; B\; \backslash otimes\_k\; B\text{\'}\; </annotation></semantics>$$

However, the tensor product of simple algebras need not be simple! And the tensor product of division algebras need not be a division algebra, or even simple. So, we have to be a bit careful if we want a workable multiplication in the Brauer monoid.

For example, take $<semantics>k=\mathbb{R}<annotation\; encoding="application/x-tex">k\; =\; \backslash mathbb\{R\}</annotation></semantics>$. The division algebras over $<semantics>\mathbb{R}<annotation\; encoding="application/x-tex">\backslash mathbb\{R\}</annotation></semantics>$ are $<semantics>\mathbb{R},\u2102<annotation\; encoding="application/x-tex">\backslash mathbb\{R\},\; \backslash mathbb\{C\}</annotation></semantics>$ and the quaternions $<semantics>\mathbb{H}<annotation\; encoding="application/x-tex">\backslash mathbb\{H\}</annotation></semantics>$. We have

$$<semantics>\mathbb{H}{\otimes}_{\mathbb{R}}\mathbb{H}\cong \mathbb{R}[4]\simeq \mathbb{R}<annotation\; encoding="application/x-tex">\; \backslash mathbb\{H\}\; \backslash otimes\_\{\backslash mathbb\{R\}\}\; \backslash mathbb\{H\}\; \backslash cong\; \backslash mathbb\{R\}[4]\; \backslash simeq\; \backslash mathbb\{R\}\; </annotation></semantics>$$

so this particular tensor product of division algebras over $<semantics>\mathbb{R}<annotation\; encoding="application/x-tex">\backslash mathbb\{R\}</annotation></semantics>$ is simple and thus Morita equivalent to another division algebra over $<semantics>\mathbb{R}<annotation\; encoding="application/x-tex">\backslash mathbb\{R\}</annotation></semantics>$. On the other hand,

$$<semantics>\u2102{\otimes}_{\mathbb{R}}\u2102\cong \u2102\oplus \u2102<annotation\; encoding="application/x-tex">\; \backslash mathbb\{C\}\; \backslash otimes\_\{\backslash mathbb\{R\}\}\; \backslash mathbb\{C\}\; \backslash cong\; \backslash mathbb\{C\}\; \backslash oplus\; \backslash mathbb\{C\}\; </annotation></semantics>$$

and this is not a division algebra, nor even simple, nor even Morita equivalent to a simple algebra.

What’s the problem with the latter example? The problem turns out to be that the division algebra $<semantics>\u2102<annotation\; encoding="application/x-tex">\backslash mathbb\{C\}</annotation></semantics>$ does not have $<semantics>\mathbb{R}<annotation\; encoding="application/x-tex">\backslash mathbb\{R\}</annotation></semantics>$ as its center: it has a larger field, namely $<semantics>\u2102<annotation\; encoding="application/x-tex">\backslash mathbb\{C\}</annotation></semantics>$ itself, as its center.

It turns out that if you tensor two simple algebras over a field $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ and they both have just $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ as their center, the result is again simple. So, in Brauer theory, people restrict attention to simple algebras over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ having just $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ as their center. These are called central simple algebras over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$. The set of Morita equivalence classes of these is closed under tensor product, so it becomes a monoid. And this monoid happens to be be an abelian group: Brauer group of $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$, denoted **$<semantics>\mathrm{Br}(k)<annotation\; encoding="application/x-tex">Br(k)</annotation></semantics>$**.
I want to work with *all* simple algebras over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$. So I will need to change this recipe a bit. But it will still be good to compute a few Brauer groups.

To do this, it pays to note that element of $<semantics>\mathrm{Br}(k)<annotation\; encoding="application/x-tex">Br(k)</annotation></semantics>$ has a representative that is a division algebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ whose center is $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$. Why? Every simple algebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ is $<semantics>D[n]<annotation\; encoding="application/x-tex">D[n]</annotation></semantics>$ for some division algebra $<semantics>D<annotation\; encoding="application/x-tex">D</annotation></semantics>$ over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$. $<semantics>D[n]<annotation\; encoding="application/x-tex">D[n]</annotation></semantics>$ is central simple over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ iff the center of $<semantics>D<annotation\; encoding="application/x-tex">D</annotation></semantics>$ is $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$, and $<semantics>D[n]<annotation\; encoding="application/x-tex">D[n]</annotation></semantics>$ is Morita equivalent to $<semantics>D<annotation\; encoding="application/x-tex">D</annotation></semantics>$. Using this, we easily see:

**Example.** The Brauer group $<semantics>\mathrm{Br}(\mathbb{R})<annotation\; encoding="application/x-tex">Br(\backslash mathbb\{R\})</annotation></semantics>$ is $<semantics>{\mathbb{Z}}_{2}<annotation\; encoding="application/x-tex">\backslash mathbb\{Z\}\_2</annotation></semantics>$, the 2-element group consisting of $<semantics>[\mathbb{R}]<annotation\; encoding="application/x-tex">[\backslash mathbb\{R\}]</annotation></semantics>$ and $<semantics>[\mathbb{H}]<annotation\; encoding="application/x-tex">[\backslash mathbb\{H\}]</annotation></semantics>$. We have

$$<semantics>[\mathbb{R}]\cdot [\mathbb{R}]=[\mathbb{R}{\otimes}_{\mathbb{R}}\mathbb{R}]=[\mathbb{R}]<annotation\; encoding="application/x-tex">\; [\backslash mathbb\{R\}]\; \backslash cdot\; [\backslash mathbb\{R\}]\; =\; [\backslash mathbb\{R\}\; \backslash otimes\_\backslash mathbb\{R\}\; \backslash mathbb\{R\}]\; =\; [\backslash mathbb\{R\}]\; </annotation></semantics>$$

$$<semantics>[\mathbb{R}]\cdot [\mathbb{H}]=[\mathbb{R}{\otimes}_{\mathbb{R}}\mathbb{H}]=[\mathbb{H}]<annotation\; encoding="application/x-tex">\; [\backslash mathbb\{R\}]\; \backslash cdot\; [\backslash mathbb\{H\}]\; =\; [\backslash mathbb\{R\}\; \backslash otimes\_\backslash mathbb\{R\}\; \backslash mathbb\{H\}]\; =\; [\backslash mathbb\{H\}]\; </annotation></semantics>$$

$$<semantics>[\mathbb{H}]\cdot [\mathbb{H}]=[\mathbb{H}{\otimes}_{\mathbb{R}}\mathbb{H}]=[\mathbb{R}]<annotation\; encoding="application/x-tex">\; [\backslash mathbb\{H\}]\; \backslash cdot\; [\backslash mathbb\{H\}]\; =\; [\backslash mathbb\{H\}\; \backslash otimes\_\backslash mathbb\{R\}\; \backslash mathbb\{H\}]\; =\; [\backslash mathbb\{R\}]\; </annotation></semantics>$$

**Example.** The Brauer group of any algebraically closed field $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ is trivial, since the only division algebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ is $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ itself. Thus $<semantics>\mathrm{Br}(\u2102)=1<annotation\; encoding="application/x-tex">Br(\backslash mathbb\{C\})\; =\; 1</annotation></semantics>$.

**Example.** The Brauer group of any finite field $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ is trivial, since the only division algebras over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ are fields that are finite extensions of $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$, and of these only $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ itself has $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ as center.

**Example.** Just so you don’t get the impression that Brauer groups tend to be boring, consider the Brauer group of the rational numbers:

$$<semantics>\mathrm{Br}(\mathbb{Q})=\{(a,x):\phantom{\rule{thickmathspace}{0ex}}a\in \{0,\frac{1}{2}\},\phantom{\rule{1em}{0ex}}x\in \underset{p}{\u2a01}\mathbb{Q}/\mathbb{Z},\phantom{\rule{1em}{0ex}}a+\sum _{p}{x}_{p}=0\}<annotation\; encoding="application/x-tex">\; Br(\backslash mathbb\{Q\})\; =\; \backslash left\backslash \{\; (a,x)\; :\; \backslash ;\; a\; \backslash in\; \backslash \{0,\backslash frac\{1\}\{2\}\backslash \},\; \backslash quad\; x\; \backslash in\; \backslash bigoplus\_p\; \backslash mathbb\{Q\}/\backslash mathbb\{Z\},\; \backslash quad\; a\; +\; \backslash sum\_p\; x\_p\; =\; 0\; \backslash right\backslash \}\; </annotation></semantics>$$

where the sum is over all primes. This is a consequence of the Albert–Brauer–Hasse–Noether theorem. The funny-looking $<semantics>\{0,\frac{1}{2}\}<annotation\; encoding="application/x-tex">\backslash \{0,\backslash frac\{1\}\{2\}\backslash \}</annotation></semantics>$ is just a way to think about the group $<semantics>{\mathbb{Z}}_{2}<annotation\; encoding="application/x-tex">\backslash mathbb\{Z\}\_2</annotation></semantics>$ as a subgroup of $<semantics>\mathbb{Q}/\mathbb{Z}<annotation\; encoding="application/x-tex">\backslash mathbb\{Q\}/\backslash mathbb\{Z\}</annotation></semantics>$. The elements of this correspond to $<semantics>\mathbb{Q}<annotation\; encoding="application/x-tex">\backslash mathbb\{Q\}</annotation></semantics>$ itself and a rational version of the quaternions. The other stuff comes from studying the situation ‘locally’ one prime at a time. However, the two aspects interact.

## The Brauer monoid of a field

Let $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ be a field and $<semantics>\overline{k}<annotation\; encoding="application/x-tex">\backslash overline\{k\}</annotation></semantics>$ its algebraic completion. Let $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$ be the set of intermediate fields

$$<semantics>k\subseteq F\subseteq \overline{k}<annotation\; encoding="application/x-tex">\; k\; \backslash subseteq\; F\; \backslash subseteq\; \backslash overline\{k\}\; </annotation></semantics>$$

where $<semantics>F<annotation\; encoding="application/x-tex">F</annotation></semantics>$ is a finite extension of $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$. This set $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$ is partially ordered by inclusion, and in fact it is a **semilattice**: any finite subset of $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$ has a least upper bound. We write $<semantics>F\vee F\prime <annotation\; encoding="application/x-tex">F\; \backslash vee\; F\text{\'}</annotation></semantics>$ for the least upper bound of $<semantics>F,F\prime \in L<annotation\; encoding="application/x-tex">F,F\text{\'}\; \backslash in\; L</annotation></semantics>$. This is just the smallest subfield of $<semantics>\overline{k}<annotation\; encoding="application/x-tex">\backslash overline\{k\}</annotation></semantics>$ containing both $<semantics>F<annotation\; encoding="application/x-tex">F</annotation></semantics>$ and $<semantics>F\prime <annotation\; encoding="application/x-tex">F\text{\'}</annotation></semantics>$.

We define the **Brauer monoid** of $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ to be the disjoint union

$$<semantics>\mathrm{BR}(k)=\coprod _{F\in L}\mathrm{Br}(F)<annotation\; encoding="application/x-tex">\; BR(k)\; =\; \backslash coprod\_\{F\; \backslash in\; L\}\; Br(F)\; </annotation></semantics>$$

So, every simple algebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ shows up in here: if $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ is a simple algebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ with center $<semantics>F<annotation\; encoding="application/x-tex">F</annotation></semantics>$, the Morita equivalence class $<semantics>[A]<annotation\; encoding="application/x-tex">[A]</annotation></semantics>$ will appear as an element of $<semantics>\mathrm{Br}(F)<annotation\; encoding="application/x-tex">Br(F)</annotation></semantics>$. However, isomorphic copies of the same simple algebra will show up repeatedly in the Brauer monoid, since we may have $<semantics>F\ne F\prime <annotation\; encoding="application/x-tex">F\; \backslash ne\; F\text{\'}</annotation></semantics>$ but still $<semantics>F\cong F\prime <annotation\; encoding="application/x-tex">F\; \backslash cong\; F\text{\'}</annotation></semantics>$.

How do we define multiplication in the Brauer monoid? The key is that the Brauer group is functorial. Suppose we have an inclusion of fields $<semantics>F\subseteq F\prime <annotation\; encoding="application/x-tex">F\; \backslash subseteq\; F\text{\'}</annotation></semantics>$ in the semilattice $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$. Then we get a homomorphism

$$<semantics>{\mathrm{Br}}_{F\prime ,F}:\mathrm{Br}(F)\to \mathrm{Br}(F\prime )<annotation\; encoding="application/x-tex">\; Br\_\{F\text{\'},\; F\}\; :\; Br(F)\; \backslash to\; Br(F\text{\'})\; </annotation></semantics>$$

as follows. Any element $<semantics>[A]\in \mathrm{Br}(F)<annotation\; encoding="application/x-tex">[A]\; \backslash in\; Br(F)</annotation></semantics>$ comes from a central simple algebra $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ over $<semantics>F<annotation\; encoding="application/x-tex">F</annotation></semantics>$; the algebra $<semantics>F\prime {\otimes}_{F}A<annotation\; encoding="application/x-tex">F\text{\'}\; \backslash otimes\_F\; A</annotation></semantics>$ will be central simple over $<semantics>F\prime <annotation\; encoding="application/x-tex">F\text{\'}</annotation></semantics>$, and we define

$$<semantics>{\mathrm{Br}}_{F\prime ,F}[A]=[F\prime {\otimes}_{F}A]<annotation\; encoding="application/x-tex">\; Br\_\{F\text{\'},F\}\; [A]\; =\; [F\text{\'}\; \backslash otimes\_F\; A]\; </annotation></semantics>$$

Of course we need to check that this is well-defined, but this is well-known. People call $<semantics>{\mathrm{Br}}_{F\prime ,F}<annotation\; encoding="application/x-tex">Br\_\{F\text{\'},F\}</annotation></semantics>$ **restriction**, since larger fields have smaller Brauer groups, but I’d prefer to call it ‘extension’, since we’re extending an algebra to be defined over a larger field.

It’s easy to see that if $<semantics>F\subseteq F\prime \subseteq F\u2033<annotation\; encoding="application/x-tex">F\; \backslash subseteq\; F\text{\'}\; \backslash subseteq\; F\text{\'}\text{\'}</annotation></semantics>$ then

$$<semantics>{\mathrm{Br}}_{F\u2033,F}={\mathrm{Br}}_{F\u2033,F\prime}{\mathrm{Br}}_{F\prime ,F}<annotation\; encoding="application/x-tex">\; Br\_\{F\text{\'}\text{\'},\; F\}\; =\; Br\_\{F\text{\'}\text{\'}\; ,F\text{\'}\}\; Br\_\{F\text{\'},\; F\}\; </annotation></semantics>$$

and this together with

$$<semantics>{\mathrm{Br}}_{F,F}={1}_{\mathrm{Br}(F)}<annotation\; encoding="application/x-tex">\; Br\_\{F,F\}\; =\; 1\_\{Br(F)\}\; </annotation></semantics>$$

implies that we have a functor

$$<semantics>\mathrm{Br}:L\to \mathrm{AbGp}<annotation\; encoding="application/x-tex">\; Br:\; L\; \backslash to\; AbGp\; </annotation></semantics>$$

So now suppose we have two elements of $<semantics>\mathrm{BR}(k)<annotation\; encoding="application/x-tex">BR(k)</annotation></semantics>$ and we want to multiply them. To do this, we simply write them as $<semantics>[A]\in \mathrm{Br}(F)<annotation\; encoding="application/x-tex">[A]\; \backslash in\; Br(F)</annotation></semantics>$ and $<semantics>[A\prime ]\in \mathrm{Br}(F\prime )<annotation\; encoding="application/x-tex">[A\text{\'}]\; \backslash in\; Br(F\text{\'})</annotation></semantics>$, map them both into $<semantics>\mathrm{Br}(F\vee F\prime )<annotation\; encoding="application/x-tex">Br(F\; \backslash vee\; F\text{\'})</annotation></semantics>$, and then multiply them there:

$$<semantics>[A]\cdot [A\prime ]\phantom{\rule{thickmathspace}{0ex}}:=\phantom{\rule{thickmathspace}{0ex}}{\mathrm{Br}}_{F\vee F\prime ,F}[A]\phantom{\rule{thickmathspace}{0ex}}\cdot {\mathrm{Br}}_{F\vee F\prime ,F\prime}[A\prime ]<annotation\; encoding="application/x-tex">\; [A]\; \backslash cdot\; [A\text{\'}]\; \backslash ;\; :=\; \backslash ;\; Br\_\{F\; \backslash vee\; F\text{\'},\; F\}\; [A]\; \backslash ;\; \backslash cdot\; Br\_\{F\; \backslash vee\; F\text{\'},\; F\text{\'}\}\; [A\text{\'}]\; </annotation></semantics>$$

This can also be expressed with less jargon as follows:

$$<semantics>[A]\cdot [A\prime ]=[A{\otimes}_{F}(F\vee F\prime ){\otimes}_{F\prime}A\prime ]<annotation\; encoding="application/x-tex">\; [A]\; \backslash cdot\; [A\text{\'}]\; =\; [A\; \backslash otimes\_F\; (F\; \backslash vee\; F\text{\'})\; \backslash otimes\_\{F\text{\'}\}\; A\text{\'}]\; </annotation></semantics>$$

However, the functorial approach gives a nice outlook on this basic result:

**Proposition.** With the above multiplication, $<semantics>\mathrm{BR}(k)<annotation\; encoding="application/x-tex">BR(k)</annotation></semantics>$ is a commutative monoid.

**Proof.** The multiplicative identity is $<semantics>[k]\in \mathrm{Br}(k)<annotation\; encoding="application/x-tex">[k]\; \backslash in\; Br(k)</annotation></semantics>$, and commutativity is obvious, so the only thing to check is associativity. This is easy enough to do directly, but it’s a bit enlightening to notice that it’s a special case of an idea that goes back to A. H. Clifford.

In modern language: suppose we have any semilattice $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$ and any functor $<semantics>B:L\to \mathrm{AbGp}<annotation\; encoding="application/x-tex">B:\; L\; \backslash to\; AbGp</annotation></semantics>$. This gives an abelian group $<semantics>B(x)<annotation\; encoding="application/x-tex">B(x)</annotation></semantics>$ for any $<semantics>x\in L<annotation\; encoding="application/x-tex">x\; \backslash in\; L</annotation></semantics>$, and a homomorphism

$$<semantics>{B}_{x\prime ,x}:B(x)\to B(x\prime )<annotation\; encoding="application/x-tex">\; B\_\{x\text{\'},\; x\}\; :\; B(x)\; \backslash to\; B(x\text{\'})\; </annotation></semantics>$$

whenever $<semantics>a\le a\prime <annotation\; encoding="application/x-tex">a\; \backslash le\; a\text{\'}</annotation></semantics>$. Then the disjoint union

$$<semantics>\coprod _{x\in L}B(x)<annotation\; encoding="application/x-tex">\; \backslash coprod\_\{x\; \backslash in\; L\}\; B(x)\; </annotation></semantics>$$

becomes a commutative monoid if we define the product of $<semantics>a\in B(x)<annotation\; encoding="application/x-tex">a\; \backslash in\; B(x)</annotation></semantics>$ and $<semantics>a\prime \in B(x\prime )<annotation\; encoding="application/x-tex">a\text{\'}\; \backslash in\; B(x\text{\'})</annotation></semantics>$ by

$$<semantics>a\cdot a\prime ={B}_{x\vee x\prime ,x}(a)\phantom{\rule{thickmathspace}{0ex}}\cdot \phantom{\rule{thickmathspace}{0ex}}{B}_{x\vee x\prime ,x\prime}(a\prime )<annotation\; encoding="application/x-tex">\; a\; \backslash cdot\; a\text{\'}\; =\; B\_\{x\; \backslash vee\; x\text{\'},x\}\; (a)\; \backslash ;\; \backslash cdot\; \backslash ;\; B\_\{x\; \backslash vee\; x\text{\'},\; x\text{\'}\}(a\text{\'})\; </annotation></semantics>$$

Checking associativity is an easy fun calculation, so I won’t deprive you of the pleasure. Moreover, there’s nothing special about abelian groups here: a functor $<semantics>B<annotation\; encoding="application/x-tex">B</annotation></semantics>$ from $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$ to commutative monoids would work just as well. ∎

Let’s see a couple of examples:

**Example.** The Brauer monoid of the real numbers is the disjoint union

$$<semantics>\mathrm{BR}(\mathbb{R})=\mathrm{Br}(\mathbb{R})\bigsqcup \mathrm{Br}(\u2102)<annotation\; encoding="application/x-tex">\; BR(\backslash mathbb\{R\})\; =\; Br(\backslash mathbb\{R\})\; \backslash sqcup\; Br(\backslash mathbb\{C\})\; </annotation></semantics>$$

This has three elements: $<semantics>[\mathbb{R}]<annotation\; encoding="application/x-tex">[\backslash mathbb\{R\}]</annotation></semantics>$, $<semantics>[\u2102]<annotation\; encoding="application/x-tex">[\backslash mathbb\{C\}]</annotation></semantics>$ and $<semantics>[\mathbb{H}]<annotation\; encoding="application/x-tex">[\backslash mathbb\{H\}]</annotation></semantics>$. Leaving out the brackets, the multiplication table is

$$<semantics>\begin{array}{lrrr}\cdot & \mathbb{R}& \u2102& \mathbb{H}\\ \mathbb{R}& \mathbb{R}& \u2102& \mathbb{H}\\ \u2102& \u2102& \u2102& \u2102\\ \mathbb{H}& \mathbb{H}& \u2102& \mathbb{R}\end{array}<annotation\; encoding="application/x-tex">\; \backslash begin\{array\}\{lrrr\}\; \backslash cdot\; \&\; \backslash mathbf\{\backslash mathbb\{R\}\}\; \&\; \backslash mathbf\{\backslash mathbb\{C\}\}\; \&\; \backslash mathbf\{\backslash mathbb\{H\}\}\; \backslash \backslash \; \backslash mathbf\{\backslash mathbb\{R\}\}\; \&\; \backslash mathbb\{R\}\; \&\; \backslash mathbb\{C\}\; \&\backslash mathbb\{H\}\; \backslash \backslash \; \backslash mathbf\{\backslash mathbb\{C\}\}\; \&\; \backslash mathbb\{C\}\; \&\; \backslash mathbb\{C\}\; \&\; \backslash mathbb\{C\}\; \backslash \backslash \; \backslash mathbf\{\backslash mathbb\{H\}\}\; \&\; \backslash mathbb\{H\}\; \&\; \backslash mathbb\{C\}\; \&\; \backslash mathbb\{R\}\; \backslash end\{array\}\; </annotation></semantics>$$

So, this monoid is isomorphic to the multiplicative monoid $<semantics>\U0001d7db=\{1,0,-1\}<annotation\; encoding="application/x-tex">\backslash mathbb\{3\}\; =\; \backslash \{1,\; 0,\; -1\backslash \}</annotation></semantics>$. This formalizes the multiplicative aspect of Dyson’s ‘threefold way’, which I started grappling with in my paper Division algebras and quantum theory. If you read that paper you can see why I care: Hilbert spaces over the real numbers, complex numbers and quaternions are all important in quantum theory, so they must fit into a single structure. The Brauer monoid is a nice way to describe this structure.

**Example.** The Brauer monoid of a finite field $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ is the disjoint union

$$<semantics>\mathrm{BR}(k)=\coprod _{F\in L}\mathrm{Br}(F)<annotation\; encoding="application/x-tex">\; BR(k)\; =\; \backslash coprod\_\{F\; \backslash in\; L\}\; Br(F)\; </annotation></semantics>$$

where $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$ is the lattice of subfields of the algebraic closure $<semantics>\overline{k}<annotation\; encoding="application/x-tex">\backslash overline\{k\}</annotation></semantics>$ that are finite extensions of $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$. However, we’ve seen that $<semantics>\mathrm{Br}(F)<annotation\; encoding="application/x-tex">Br(F)</annotation></semantics>$ is always the trivial group. Thus

$$<semantics>\mathrm{Br}(k)\cong L<annotation\; encoding="application/x-tex">\; Br(k)\; \backslash cong\; L\; </annotation></semantics>$$

with the monoid structure being the operation $<semantics>\vee <annotation\; encoding="application/x-tex">\backslash vee</annotation></semantics>$ in the lattice $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$.

**Example.** The Brauer monoid of $<semantics>\mathbb{Q}<annotation\; encoding="application/x-tex">\backslash mathbb\{Q\}</annotation></semantics>$ seems quite complicated to me, since it’s the disjoint union of $<semantics>\mathrm{Br}(F)<annotation\; encoding="application/x-tex">Br(F)</annotation></semantics>$ for all $<semantics>F\subset \overline{\mathbb{Q}}<annotation\; encoding="application/x-tex">F\; \backslash subset\; \backslash overline\{\backslash mathbb\{Q\}\}</annotation></semantics>$ that are finite extensions of $<semantics>\mathbb{Q}<annotation\; encoding="application/x-tex">\backslash mathbb\{Q\}</annotation></semantics>$. Such fields $<semantics>F<annotation\; encoding="application/x-tex">F</annotation></semantics>$ are called **algebraic number fields**, and their Brauer groups can, I believe, be computed using the Albert–Brauer–Hasse–Noether theorem. However, here we are doing this for *all* algebraic number fields, and also keeping track of how they ‘fit together’ using the so-called restriction maps $<semantics>{\mathrm{Br}}_{F\prime ,F}:\mathrm{Br}(F)\to \mathrm{Br}(F\prime )<annotation\; encoding="application/x-tex">Br\_\{F\text{\'},\; F\}\; :\; Br(F)\; \backslash to\; Br(F\text{\'})</annotation></semantics>$ whenever $<semantics>F\subseteq F\prime <annotation\; encoding="application/x-tex">F\backslash subseteq\; F\text{\'}</annotation></semantics>$. The absolute Galois group of a field always acts on its Brauer monoid, so the rather fearsome absolute Galois group of $<semantics>\mathbb{Q}<annotation\; encoding="application/x-tex">\backslash mathbb\{Q\}</annotation></semantics>$ acts on $<semantics>\mathrm{Br}(\mathbb{Q})<annotation\; encoding="application/x-tex">Br(\backslash mathbb\{Q\})</annotation></semantics>$, for whatever that’s worth.

Fleeing the siren song of number theory, let us move on to my main topic of interest, which is the ‘super’ or $<semantics>{\mathbb{Z}}_{2}<annotation\; encoding="application/x-tex">\backslash mathbb\{Z\}\_2</annotation></semantics>$-graded version of this whole story.

## Review of superalgebra

We now want to repeat everything we just did, systematically replacing the category of vector spaces over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ by the category of **super vector spaces** over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$, which are $<semantics>{\mathbb{Z}}_{2}<annotation\; encoding="application/x-tex">\backslash mathbb\{Z\}\_2</annotation></semantics>$ graded vector spaces:

$$<semantics>V={V}_{0}\oplus {V}_{1}<annotation\; encoding="application/x-tex">\; V\; =\; V\_0\; \backslash oplus\; V\_1\; </annotation></semantics>$$

We call the elements of $<semantics>{V}_{0}<annotation\; encoding="application/x-tex">V\_0</annotation></semantics>$ **even** and the elements of $<semantics>{V}_{1}<annotation\; encoding="application/x-tex">V\_1</annotation></semantics>$ **odd**. Elements of either $<semantics>{V}_{0}<annotation\; encoding="application/x-tex">V\_0</annotation></semantics>$ or $<semantics>{V}_{1}<annotation\; encoding="application/x-tex">V\_1</annotation></semantics>$ are called **homogeneous**, and we say an element $<semantics>a\in {V}_{i}<annotation\; encoding="application/x-tex">a\; \backslash in\; V\_i</annotation></semantics>$ has **degree** $<semantics>i<annotation\; encoding="application/x-tex">i</annotation></semantics>$. A morphism in the category of super vector spaces is a linear map that preserves the degree of homogeneous elements.

The category of super vector spaces is symmetric monoidal in a way where we introduce a minus sign when we switch two odd elements.

### Simple superalgebras and division superalgebras

A **superalgebra** is a monoid in the category of super vector spaces. In other words, it is a super vector space $<semantics>A={A}_{0}\oplus {A}_{1}<annotation\; encoding="application/x-tex">A\; =\; A\_0\; \backslash oplus\; A\_1</annotation></semantics>$ where the vector space $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ is an algebra in the usual sense and

$$<semantics>a\in {A}_{i},\phantom{\rule{thickmathspace}{0ex}}b\in {A}_{j}\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}a\cdot b\in {A}_{i+j}<annotation\; encoding="application/x-tex">\; a\; \backslash in\; A\_i,\; \backslash ;\; b\; \backslash in\; A\_j\; \backslash quad\; \backslash implies\; \backslash quad\; a\; \backslash cdot\; b\; \backslash in\; A\_\{i\; +\; j\}\; </annotation></semantics>$$

where we do our addition mod 2. There is a tensor product of superalgebras, where

$$<semantics>(A\otimes B{)}_{i}=\underset{i=j+k}{\u2a01}{A}_{j}\otimes {B}_{k}<annotation\; encoding="application/x-tex">\; (A\; \backslash otimes\; B)\_i\; =\; \backslash bigoplus\_\{i\; =\; j\; +\; k\}\; A\_j\; \backslash otimes\; B\_k\; </annotation></semantics>$$

and multiplication is defined on homogeneous elements by:

$$<semantics>(a\otimes b)(a\prime \otimes b\prime )=(-1{)}^{i+j}\phantom{\rule{thickmathspace}{0ex}}aa\prime \otimes bb\prime <annotation\; encoding="application/x-tex">\; (a\; \backslash otimes\; b)(a\text{\'}\; \backslash otimes\; b\text{\'})\; =\; (-1)^\{i+j\}\; \backslash ;\; a\; a\text{\'}\; \backslash otimes\; b\; b\text{\'}\; </annotation></semantics>$$

where $<semantics>b\in {B}_{i},a\prime \in {A}_{j}<annotation\; encoding="application/x-tex">b\; \backslash in\; B\_i,\; a\text{\'}\; \backslash in\; A\_j</annotation></semantics>$ are the elements getting switched.

An ideal $<semantics>I<annotation\; encoding="application/x-tex">I</annotation></semantics>$ of a superalgebra is **homogeneous** if it is of the form

$$<semantics>I={I}_{0}\oplus {I}_{1}<annotation\; encoding="application/x-tex">\; I\; =\; I\_0\; \backslash oplus\; I\_1\; </annotation></semantics>$$

where $<semantics>{I}_{i}\subseteq {A}_{i}<annotation\; encoding="application/x-tex">I\_i\; \backslash subseteq\; A\_i</annotation></semantics>$. We can take the quotient of a superalgebra by a homogeneous two-sided ideal and get another superalgebra. So, we say a superalgebra $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ over $<semantics>F<annotation\; encoding="application/x-tex">F</annotation></semantics>$ is **simple** if its only two-sided homogeneous ideals are $<semantics>\{0\}<annotation\; encoding="application/x-tex">\backslash \{0\backslash \}</annotation></semantics>$ and $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ itself.

A **division superalgebra** over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ is a superalgebra $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ such that if $<semantics>a\ne 0<annotation\; encoding="application/x-tex">a\; \backslash ne\; 0</annotation></semantics>$ is homogeneous then there exists $<semantics>b\in A<annotation\; encoding="application/x-tex">b\; \backslash in\; A</annotation></semantics>$ such that $<semantics>ab=ba=1<annotation\; encoding="application/x-tex">a\; b=\; b\; a\; =\; 1</annotation></semantics>$.

At this point it is clear what we aim to do: generalize Brauer groups to this ‘super’ context by replacing division algebras with division superalgebras. Luckily this was already done a long time ago, by Wall:

• C. T. C. Wall, Graded Brauer groups, *Journal für die reine und angewandte Mathematik* **213** (1963–1964), 187–199.

He showed there are 10 division superalgebras over $<semantics>\mathbb{R}<annotation\; encoding="application/x-tex">\backslash mathbb\{R\}</annotation></semantics>$ and showed how 8 of these become elements of a kind of super Brauer group for $<semantics>\mathbb{R}<annotation\; encoding="application/x-tex">\backslash mathbb\{R\}</annotation></semantics>$, now called the ‘Brauer–Wall’ group. The other 2 become elements of the Brauer–Wall group of $<semantics>\u2102<annotation\; encoding="application/x-tex">\backslash mathbb\{C\}</annotation></semantics>$. A more up-to-date treatment of *some* of this material can be found here:

• Pierre Deligne, Notes on spinors, in *Quantum Fields and Strings: a Course for Mathematicians*, vol. 1, AMS. Providence, RI, 1999, pp. 99–135.

Nontrivial results that I state without proof will come from these sources.

Every division superalgebra is simple. Conversely, we want a super-Wedderburn theorem describing simple superalgebras in terms of division superalgebras. However, this must be more complicated than the ordinary Wedderburn theorem saying every simple algebra is a matrix algebra $<semantics>D[n]<annotation\; encoding="application/x-tex">D[n]</annotation></semantics>$ with $<semantics>D<annotation\; encoding="application/x-tex">D</annotation></semantics>$ a division algebra.

After all, besides matrix algebras, we have ‘matrix superalgebras’ to contend with. For any $<semantics>p,q\ge 0<annotation\; encoding="application/x-tex">p,q\; \backslash ge\; 0</annotation></semantics>$ let $<semantics>{k}^{p|q}<annotation\; encoding="application/x-tex">k^\{p|q\}</annotation></semantics>$ be the super vector space with even part $<semantics>{k}^{p}<annotation\; encoding="application/x-tex">k^p</annotation></semantics>$ and odd part $<semantics>{k}^{q}<annotation\; encoding="application/x-tex">k^q</annotation></semantics>$. Then its endomorphism algebra

$$<semantics>k[p|q]=\mathrm{End}({k}^{p|q})<annotation\; encoding="application/x-tex">\; k[p|q]\; =\; End(k^\{p|q\})</annotation></semantics>$$

becomes a superalgebra in a standard way, called a **matrix superalgebra**. Matrix superalgebras are always simple.

Deligne gives a classification of ‘central simple’ superalgebras, and from this we can derive a super-Wedderburn theorem. But what does ‘central simple’ mean in this context?

The **supercommutator** of two homogeneous elements $<semantics>a\in {A}_{i}<annotation\; encoding="application/x-tex">a\; \backslash in\; A\_i</annotation></semantics>$, $<semantics>b\in {A}_{j}<annotation\; encoding="application/x-tex">b\; \backslash in\; A\_j</annotation></semantics>$ of a superalgebra $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ is

$$<semantics>[a,b]=ab-(-1{)}^{i+j}ba<annotation\; encoding="application/x-tex">\; [a,b]\; =\; a\; b\; -\; (-1)^\{i+j\}\; b\; a</annotation></semantics>$$

We can extend this by bilinearity to all elements of $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$. We say $<semantics>a,b\in A<annotation\; encoding="application/x-tex">a,b\; \backslash in\; A</annotation></semantics>$ **supercommute** if $<semantics>[a,b]=0<annotation\; encoding="application/x-tex">[a,b]=\; 0</annotation></semantics>$. The **supercenter** of $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ is the set of elements in $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ that supercommute with every element of $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$. If all elements of $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ supercommute, or equivalently if the supercenter of $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ is all of $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$, we say $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ is **supercommutative**.

I believe a superalgebra $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ is **central simple** if $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ is simple and its supercenter is just $<semantics>k\subseteq {A}_{0}<annotation\; encoding="application/x-tex">k\; \backslash subseteq\; A\_0</annotation></semantics>$, the scalar multiples of the identity. Deligne gives a more complicated definition of ‘central simple’, but then in Remark 3.5 proves it is equivalent to being semisimple with supercenter just $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$. I believe this is equivalent to the more reasonable-sounding condition I just gave, but have not carefully checked.

In Remark 3.5, Deligne says that by copying an argument in Chapter 8 of Bourbaki’s *Algebra* one can show:

**Proposition.** Any central simple superalgebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ is of the form
$<semantics>D[p|q]<annotation\; encoding="application/x-tex">D[p|q]</annotation></semantics>$ for some division superalgebra $<semantics>D<annotation\; encoding="application/x-tex">D</annotation></semantics>$ whose supercenter is $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$. Conversely, any superalgebra of this form is central simple.

Starting from this, Guo Chuan Thiang showed me how to prove the:

**Super-Wedderburn Theorem.** Suppose $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ is a simple superalgebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$, where $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ is a field not of characteristic 2. Its supercenter $<semantics>Z(A)<annotation\; encoding="application/x-tex">Z(A)</annotation></semantics>$ is purely even, and $<semantics>Z(A)<annotation\; encoding="application/x-tex">Z(A)</annotation></semantics>$ is a field extending $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$. $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ is isomorphic to $<semantics>D[p|q]<annotation\; encoding="application/x-tex">D[p|q]</annotation></semantics>$ where $<semantics>D<annotation\; encoding="application/x-tex">D</annotation></semantics>$ is some division superalgebra $<semantics>D<annotation\; encoding="application/x-tex">D</annotation></semantics>$ over $<semantics>Z(A)<annotation\; encoding="application/x-tex">Z(A)</annotation></semantics>$.

It follows that any simple superalgebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ is of the form $<semantics>D[p|q]<annotation\; encoding="application/x-tex">D[p|q]</annotation></semantics>$ where $<semantics>D<annotation\; encoding="application/x-tex">D</annotation></semantics>$ is a division superalgebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$. Conversely, if $<semantics>D<annotation\; encoding="application/x-tex">D</annotation></semantics>$ is any division algebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$, then $<semantics>D[p,q]<annotation\; encoding="application/x-tex">D[p,q]</annotation></semantics>$ is a simple superalgebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$.

**Proof.** Suppose $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ is a simple superalgebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$, and let $<semantics>Z(A)<annotation\; encoding="application/x-tex">Z(A)</annotation></semantics>$ be its supercenter. Suppose $<semantics>a<annotation\; encoding="application/x-tex">a</annotation></semantics>$ is a nonzero homogeneous element of $<semantics>Z(A)<annotation\; encoding="application/x-tex">Z(A)</annotation></semantics>$. Then $<semantics>aA<annotation\; encoding="application/x-tex">a\; A</annotation></semantics>$ is a graded two-sided ideal of $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$. Since this ideal contains $<semantics>a<annotation\; encoding="application/x-tex">a</annotation></semantics>$ it is nonzero. Thus, this must be $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ itself. So, there exists $<semantics>b\in A<annotation\; encoding="application/x-tex">b\; \backslash in\; A</annotation></semantics>$ such that $<semantics>ab=1<annotation\; encoding="application/x-tex">a\; b\; =\; 1</annotation></semantics>$.

If $<semantics>a<annotation\; encoding="application/x-tex">a</annotation></semantics>$ is even, $<semantics>b<annotation\; encoding="application/x-tex">b</annotation></semantics>$ must be as well, and we obtain $<semantics>ba=ab=1<annotation\; encoding="application/x-tex">b\; a\; =\; a\; b\; =\; 1</annotation></semantics>$, so $<semantics>a<annotation\; encoding="application/x-tex">a</annotation></semantics>$ has an inverse. Thus, the even part of $<semantics>Z(A)<annotation\; encoding="application/x-tex">Z(A)</annotation></semantics>$ is a field.

If $<semantics>a<annotation\; encoding="application/x-tex">a</annotation></semantics>$ is odd, it satisfies $<semantics>{a}^{2}=-{a}^{2}<annotation\; encoding="application/x-tex">a^2=-a^2</annotation></semantics>$. Multiplying on the left by $<semantics>b<annotation\; encoding="application/x-tex">b</annotation></semantics>$, then it follows that $<semantics>a=-a<annotation\; encoding="application/x-tex">a\; =\; -a</annotation></semantics>$, so $<semantics>a=0<annotation\; encoding="application/x-tex">a\; =\; 0</annotation></semantics>$, since $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ is not of characteristic 2.

In short, nonzero elements of $<semantics>Z(A)<annotation\; encoding="application/x-tex">Z(A)</annotation></semantics>$ must be even and invertible. It follows that $<semantics>Z(A)<annotation\; encoding="application/x-tex">Z(A)</annotation></semantics>$ is purely even, and is a field extending $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$. $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ is central over this field $<semantics>Z(A)<annotation\; encoding="application/x-tex">Z(A)</annotation></semantics>$, so by the previous proposition we see $<semantics>A\cong D[p|q]<annotation\; encoding="application/x-tex">A\; \backslash cong\; D[p|q]</annotation></semantics>$ for some division superalgebra $<semantics>D<annotation\; encoding="application/x-tex">D</annotation></semantics>$ over $<semantics>Z(A)<annotation\; encoding="application/x-tex">Z(A)</annotation></semantics>$. $<semantics>D<annotation\; encoding="application/x-tex">D</annotation></semantics>$ will automatically be a division superalgebra over the smaller field $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ as well.

Conversely, suppose $<semantics>D<annotation\; encoding="application/x-tex">D</annotation></semantics>$ is a division algebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$. Since $<semantics>D<annotation\; encoding="application/x-tex">D</annotation></semantics>$ is simple, its supercenter will be a field $<semantics>F<annotation\; encoding="application/x-tex">F</annotation></semantics>$ extending $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$. By the previous proposition $<semantics>D[p|q]<annotation\; encoding="application/x-tex">D[p|q]</annotation></semantics>$ will be a central simple superalgebra over $<semantics>F<annotation\; encoding="application/x-tex">F</annotation></semantics>$. It follows that $<semantics>D[p|q]<annotation\; encoding="application/x-tex">D[p|q]</annotation></semantics>$ is simple as a superalgebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$. ∎

Here is an all-important example:

**Example.** Let $<semantics>k[\sqrt{-1}]<annotation\; encoding="application/x-tex">k[\backslash sqrt\{-1\}]</annotation></semantics>$ be the free superalgebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ on an odd generator whose square is -1. This superalgebra has a 1-dimensional even part and a 1-dimensional odd part. It is a division superalgebra. It is not supercommutative, since $<semantics>\sqrt{-1}<annotation\; encoding="application/x-tex">\backslash sqrt\{-1\}</annotation></semantics>$ does not supercommute with itself. It is central simple: its supercenter is just $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$. Over an algebraically closed field $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ of characteristic other than 2, the only division superalgebras are $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ itself and $<semantics>k[\sqrt{-1}]<annotation\; encoding="application/x-tex">k[\backslash sqrt\{-1\}]</annotation></semantics>$.

I don’t understand what happens in characteristic 2.

### Morita equivalence and the Brauer–Wall group

The Brauer–Wall group consists of Morita equivalence classes of central simple superalgebras, or equivalently, Morita equivalence classes of division superalgebras. For this to make sense, first we need to define Morita equivalence.

Given a superalgebra $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$ over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ we define a **left module** to be a super vector space $<semantics>V<annotation\; encoding="application/x-tex">V</annotation></semantics>$ over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ equipped with a morphism (that is, a grade-preserving linear map)

$$<semantics>A\otimes V\to V<annotation\; encoding="application/x-tex">\; A\; \backslash otimes\; V\; \backslash to\; V\; </annotation></semantics>$$

obeying the usual axioms of a left module. We define a morphism of left $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$-modules in the obvious way, and let **$<semantics>\mathrm{Rep}(A)<annotation\; encoding="application/x-tex">Rep(A)</annotation></semantics>$** be the category of left $<semantics>A<annotation\; encoding="application/x-tex">A</annotation></semantics>$-modules.

We say two algebras $<semantics>A,B<annotation\; encoding="application/x-tex">A,\; B</annotation></semantics>$ over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ are **Morita equivalent** if $<semantics>\mathrm{Rep}(A)\simeq \mathrm{Rep}(B)<annotation\; encoding="application/x-tex">Rep(A)\; \backslash simeq\; Rep(B)</annotation></semantics>$. In this situation we write **$<semantics>A\simeq B<annotation\; encoding="application/x-tex">A\; \backslash simeq\; B</annotation></semantics>$**.

**Example.** Every matrix superalgebra $<semantics>k[p|q]<annotation\; encoding="application/x-tex">k[p|q]</annotation></semantics>$ is Morita equivalent to $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$.

**Example.** If $<semantics>A\simeq A\prime <annotation\; encoding="application/x-tex">A\; \backslash simeq\; A\text{\'}</annotation></semantics>$ and $<semantics>B\simeq B\prime <annotation\; encoding="application/x-tex">B\; \backslash simeq\; B\text{\'}</annotation></semantics>$ then $<semantics>A{\otimes}_{k}A\prime \simeq B{\otimes}_{k}B\prime <annotation\; encoding="application/x-tex">A\; \backslash otimes\_k\; A\text{\'}\; \backslash simeq\; B\; \backslash otimes\_k\; B\text{\'}\; </annotation></semantics>$.

**Example.** Since every central simple superalgebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ is of the form $<semantics>D[p|q]=D\otimes k[p|q]<annotation\; encoding="application/x-tex">D[p|q]\; =\; D\; \backslash otimes\; k[p|q]</annotation></semantics>$ for some division superalgebra $<semantics>D<annotation\; encoding="application/x-tex">D</annotation></semantics>$ whose supercenter is just $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$, the previous two examples imply that every central simple superalgebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ is Morita equivalent to a division superalgebra whose center is just $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$.

We define the **Brauer–Wall group** $<semantics>\mathrm{Bw}(k)<annotation\; encoding="application/x-tex">Bw(k)</annotation></semantics>$ of the field $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ to be the set of Morita equivalence classes of central simple superalgebras over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$, given the following multiplication:

$$<semantics>[A]\otimes [B]\phantom{\rule{thickmathspace}{0ex}}:=\phantom{\rule{thickmathspace}{0ex}}[A\otimes B]<annotation\; encoding="application/x-tex">\; [A]\; \backslash otimes\; [B]\; \backslash ;\; :=\; \backslash ;\; [A\; \backslash otimes\; B]\; </annotation></semantics>$$

This is well-defined because the tensor product of central simple superalgebras is again central simple. Given that, $<semantics>\mathrm{Bw}(k)<annotation\; encoding="application/x-tex">Bw(k)</annotation></semantics>$ is clearly a commutative monoid. But in fact it’s an abelian group.

Since every central simple superalgebra over $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ is Morita equivalent to a division superalgebra whose center is just $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$, we can compute Brauer–Wall groups by focusing on these division superalgebras.

**Example.** For any algebraically closed field $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$, the Brauer–Wall group $<semantics>\mathrm{Bw}(k)<annotation\; encoding="application/x-tex">Bw(k)</annotation></semantics>$ is $<semantics>{\mathbb{Z}}_{2}<annotation\; encoding="application/x-tex">\backslash mathbb\{Z\}\_2</annotation></semantics>$, where the two elements are $<semantics>[k]<annotation\; encoding="application/x-tex">[k]</annotation></semantics>$ and $<semantics>[k[\sqrt{-1}]]<annotation\; encoding="application/x-tex">[k[\backslash sqrt\{-1\}]]</annotation></semantics>$. In particular, $<semantics>\mathrm{Bw}(\u2102)<annotation\; encoding="application/x-tex">Bw(\backslash mathbb\{C\})</annotation></semantics>$ is $<semantics>{\mathbb{Z}}_{2}<annotation\; encoding="application/x-tex">\backslash mathbb\{Z\}\_2</annotation></semantics>$. Wall showed that this $<semantics>{\mathbb{Z}}_{2}<annotation\; encoding="application/x-tex">\backslash mathbb\{Z\}\_2</annotation></semantics>$ is related to the period-2 phenomenon in complex K-theory and the theory of complex Clifford algebras. The point is that

$$<semantics>\u2102[\sqrt{-1}{]}^{\otimes n}\cong \u2102{\mathrm{liff}}_{n}<annotation\; encoding="application/x-tex">\; \backslash mathbb\{C\}[\backslash sqrt\{-1\}]^\{\backslash otimes\; n\}\; \backslash cong\; \backslash mathbb\{C\}liff\_n\; </annotation></semantics>$$

where $<semantics>\u2102{\mathrm{liff}}_{n}<annotation\; encoding="application/x-tex">\backslash mathbb\{C\}liff\_n</annotation></semantics>$ is the complex Clifford algebra on $<semantics>n<annotation\; encoding="application/x-tex">n</annotation></semantics>$ square roots of -1, made into a superalgebra in the usual way. It is well-known that

$$<semantics>\u2102{\mathrm{liff}}_{2}\simeq \u2102{\mathrm{liff}}_{0}<annotation\; encoding="application/x-tex">\; \backslash mathbb\{C\}liff\_2\; \backslash simeq\; \backslash mathbb\{C\}liff\_0\; </annotation></semantics>$$

and this gives the period-2 phenomenon.

**Example.** $<semantics>\mathrm{Bw}(\mathbb{R})<annotation\; encoding="application/x-tex">Bw(\backslash mathbb\{R\})</annotation></semantics>$ is much more interesting: by a theorem of Wall, this is $<semantics>{\mathbb{Z}}_{8}<annotation\; encoding="application/x-tex">\backslash mathbb\{Z\}\_8</annotation></semantics>$. This is generated by $<semantics>[\mathbb{R}[\sqrt{-1}]]<annotation\; encoding="application/x-tex">[\backslash mathbb\{R\}[\backslash sqrt\{-1\}]]</annotation></semantics>$. Wall showed that this $<semantics>{\mathbb{Z}}_{8}<annotation\; encoding="application/x-tex">\backslash mathbb\{Z\}\_8</annotation></semantics>$ is related to the period-8 phenomenon in real K-theory and the theory of real Clifford algebras. The point is that

$$<semantics>\mathbb{R}[\sqrt{-1}{]}^{\otimes n}\cong {\mathrm{Cliff}}_{n}<annotation\; encoding="application/x-tex">\; \backslash mathbb\{R\}[\backslash sqrt\{-1\}]^\{\backslash otimes\; n\}\; \backslash cong\; Cliff\_\{n\}\; </annotation></semantics>$$

where $<semantics>{\mathrm{Cliff}}_{n}<annotation\; encoding="application/x-tex">Cliff\_\{n\}</annotation></semantics>$ is the real Clifford algebra on $<semantics>n<annotation\; encoding="application/x-tex">n</annotation></semantics>$ square roots of -1, made into a superalgebra in the usual way. It is well-known that

$$<semantics>{\mathrm{Cliff}}_{8}\simeq {\mathrm{Cliff}}_{0}<annotation\; encoding="application/x-tex">\; Cliff\_\{8\}\; \backslash simeq\; Cliff\_0\; </annotation></semantics>$$

and this gives the period-8 phenomenon.

**Example.** More generally, Wall showed that as long as $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ doesn’t have characteristic 2, $<semantics>\mathrm{Bw}(k)<annotation\; encoding="application/x-tex">Bw(k)</annotation></semantics>$ is an iterated extension of $<semantics>{\mathbb{Z}}_{2}<annotation\; encoding="application/x-tex">\backslash mathbb\{Z\}\_2</annotation></semantics>$ by $<semantics>{k}^{*}/({k}^{*}{)}^{2}<annotation\; encoding="application/x-tex">k^*/(k^*)^2</annotation></semantics>$ by $<semantics>\mathrm{Br}(k)<annotation\; encoding="application/x-tex">Br(k)</annotation></semantics>$. For a quick modern proof, see Lemma 3.7 in Deligne’s paper. In the case $<semantics>k=\mathbb{R}<annotation\; encoding="application/x-tex">k\; =\; \backslash mathbb\{R\}</annotation></semantics>$ all three of these groups are $<semantics>{\mathbb{Z}}_{2}<annotation\; encoding="application/x-tex">\backslash mathbb\{Z\}\_2</annotation></semantics>$ and the iterated extension gives $<semantics>{\mathbb{Z}}_{8}<annotation\; encoding="application/x-tex">\backslash mathbb\{Z\}\_8</annotation></semantics>$.

## The Brauer–Wall monoid

And now the rest practically writes itself. Let $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ be a field and $<semantics>\overline{k}<annotation\; encoding="application/x-tex">\backslash overline\{k\}</annotation></semantics>$ its algebraic completion. As before, let $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$ be the semilattice of intermediate fields

$$<semantics>k\subseteq F\subseteq \overline{k}<annotation\; encoding="application/x-tex">\; k\; \backslash subseteq\; F\; \backslash subseteq\; \backslash overline\{k\}\; </annotation></semantics>$$

where $<semantics>F<annotation\; encoding="application/x-tex">F</annotation></semantics>$ is a finite extension of $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$.

We define the underlying set of **Brauer–Wall monoid** of $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ to be the disjoint union

$$<semantics>\mathrm{BW}(k)=\coprod _{F\in L}\mathrm{Bw}(F)<annotation\; encoding="application/x-tex">\; BW(k)\; =\; \backslash coprod\_\{F\; \backslash in\; L\}\; Bw(F)\; </annotation></semantics>$$

To make this into a commutative monoid, we use the functoriality of the Brauer–Wall group. Suppose we have an inclusion of fields $<semantics>F\subseteq F\prime <annotation\; encoding="application/x-tex">F\; \backslash subseteq\; F\text{\'}</annotation></semantics>$ in the semilattice $<semantics>L<annotation\; encoding="application/x-tex">L</annotation></semantics>$. Then we get a homomorphism

$$<semantics>{\mathrm{Bw}}_{F\prime ,F}:\mathrm{Bw}(F)\to \mathrm{Bw}(F\prime )<annotation\; encoding="application/x-tex">\; Bw\_\{F\text{\'},\; F\}\; :\; Bw(F)\; \backslash to\; Bw(F\text{\'})\; </annotation></semantics>$$

as follows:

$$<semantics>{\mathrm{Bw}}_{F\prime ,F}[A]=[F\prime {\otimes}_{F}A]<annotation\; encoding="application/x-tex">\; Bw\_\{F\text{\'},F\}\; [A]\; =\; [F\text{\'}\; \backslash otimes\_F\; A]\; </annotation></semantics>$$

and this gives a functor

$$<semantics>\mathrm{Bw}:L\to \mathrm{AbGp}<annotation\; encoding="application/x-tex">\; Bw:\; L\; \backslash to\; AbGp\; </annotation></semantics>$$

Using this, we multiply two elements in the Brauer–Wall monoid as follows. Given $<semantics>[A]\in \mathrm{Bw}(F)<annotation\; encoding="application/x-tex">[A]\; \backslash in\; Bw(F)</annotation></semantics>$ and $<semantics>[A\prime ]\in \mathrm{Bw}(F\prime )<annotation\; encoding="application/x-tex">[A\text{\'}]\; \backslash in\; Bw(F\text{\'})</annotation></semantics>$, their product is

$$<semantics>[A]\cdot [A\prime ]\phantom{\rule{thickmathspace}{0ex}}:=\phantom{\rule{thickmathspace}{0ex}}{\mathrm{Bw}}_{F\vee F\prime ,F}[A]\phantom{\rule{thickmathspace}{0ex}}\cdot {\mathrm{Bw}}_{F\vee F\prime ,F\prime}[A\prime ]<annotation\; encoding="application/x-tex">\; [A]\; \backslash cdot\; [A\text{\'}]\; \backslash ;\; :=\; \backslash ;\; Bw\_\{F\; \backslash vee\; F\text{\'},\; F\}\; [A]\; \backslash ;\; \backslash cdot\; Bw\_\{F\; \backslash vee\; F\text{\'},\; F\text{\'}\}\; [A\text{\'}]\; </annotation></semantics>$$

or in other words

$$<semantics>[A]\cdot [A\prime ]=[A{\otimes}_{F}(F\vee F\prime ){\otimes}_{F\prime}A\prime ]<annotation\; encoding="application/x-tex">\; [A]\; \backslash cdot\; [A\text{\'}]\; =\; [A\; \backslash otimes\_F\; (F\; \backslash vee\; F\text{\'})\; \backslash otimes\_\{F\text{\'}\}\; A\text{\'}]\; </annotation></semantics>$$

**Proposition.** With the above multiplication, $<semantics>\mathrm{BR}(k)<annotation\; encoding="application/x-tex">BR(k)</annotation></semantics>$ is a commutative monoid.

**Proof.** The same argument that let us show associativity for multiplication in the Brauer monoid works again here. ∎

**Example.** As a set, the Brauer–Wall monoid of the real numbers is the disjoint union

$$<semantics>\mathrm{BW}(\mathbb{R})=\mathrm{Bw}(\mathbb{R})\bigsqcup \mathrm{Bw}(\u2102)\cong {\mathbb{Z}}_{8}\bigsqcup {\mathbb{Z}}_{2}<annotation\; encoding="application/x-tex">\; BW(\backslash mathbb\{R\})\; =\; Bw(\backslash mathbb\{R\})\; \backslash sqcup\; Bw(\backslash mathbb\{C\})\; \backslash cong\; \backslash mathbb\{Z\}\_8\; \backslash sqcup\; \backslash mathbb\{Z\}\_2\; </annotation></semantics>$$

The monoid operation — let’s call it addition now — is the usual addition on $<semantics>{\mathbb{Z}}_{8}<annotation\; encoding="application/x-tex">\backslash mathbb\{Z\}\_8</annotation></semantics>$ when applied to two elements of that group, and the usual addition on $<semantics>{\mathbb{Z}}_{2}<annotation\; encoding="application/x-tex">\backslash mathbb\{Z\}\_2</annotation></semantics>$ when applied to two elements of *that* group. The only interesting part is when we add an element $<semantics>a\in {\mathbb{Z}}_{8}<annotation\; encoding="application/x-tex">a\; \backslash in\; \backslash mathbb\{Z\}\_8</annotation></semantics>$ and an element $<semantics>b\in {\mathbb{Z}}_{2}<annotation\; encoding="application/x-tex">b\; \backslash in\; \backslash mathbb\{Z\}\_2</annotation></semantics>$. For this we need to convert $<semantics>a<annotation\; encoding="application/x-tex">a</annotation></semantics>$ into an element of $<semantics>{\mathbb{Z}}_{2}<annotation\; encoding="application/x-tex">\backslash mathbb\{Z\}\_2</annotation></semantics>$. For that we use the homomorphism

$$<semantics>{\mathrm{Bw}}_{\u2102,\mathbb{R}}:\mathrm{Bw}(\mathbb{R})\to \mathrm{Bw}(\u2102)<annotation\; encoding="application/x-tex">\; Bw\_\{\backslash mathbb\{C\},\; \backslash mathbb\{R\}\}\; :\; Bw(\backslash mathbb\{R\})\; \backslash to\; Bw(\backslash mathbb\{C\})\; </annotation></semantics>$$

which sends $<semantics>[\mathbb{R}[\sqrt{-1}]]<annotation\; encoding="application/x-tex">[\backslash mathbb\{R\}[\backslash sqrt\{-1\}]]</annotation></semantics>$ to $<semantics>[\u2102[\sqrt{-1}]]<annotation\; encoding="application/x-tex">[\backslash mathbb\{C\}[\backslash sqrt\{-1\}]]</annotation></semantics>$. More concretely,

$$<semantics>{\mathrm{Bw}}_{\u2102,\mathbb{R}}:{\mathbb{Z}}_{8}\to {\mathbb{Z}}_{2}<annotation\; encoding="application/x-tex">\; Bw\_\{\backslash mathbb\{C\},\; \backslash mathbb\{R\}\}\; :\; \backslash mathbb\{Z\}\_8\; \backslash to\; \backslash mathbb\{Z\}\_2\; </annotation></semantics>$$

takes an integer mod 8 and gives the corresponding integer mod 2.

So, very concretely,

$$<semantics>\mathrm{BW}(\mathbb{R})\cong \mathrm{\U0001d7d9\U0001d7d8}=\{0,1,2,3,4,5,6,7,0,1\}<annotation\; encoding="application/x-tex">\; BW(\backslash mathbb\{R\})\; \backslash cong\; \backslash mathbb\{10\}\; =\; \backslash \{0,1,2,3,4,5,6,7,\backslash mathbf\{0\},\; \backslash mathbf\{1\}\backslash \}</annotation></semantics>$$

where the monoid operation in $<semantics>\mathrm{\U0001d7d9\U0001d7d8}<annotation\; encoding="application/x-tex">\backslash mathbb\{10\}</annotation></semantics>$ is addition mod 8 for two lightface numbers, but addition mod 2 for two boldface numbers or a boldface and a lightface one.

## Conclusion

I had meant to include a section explaining in detail how the 10 elements of this monoid $<semantics>\mathrm{\U0001d7d9\U0001d7d8}<annotation\; encoding="application/x-tex">\backslash mathbb\{10\}</annotation></semantics>$ correspond to 10 kinds of matter, but this post is getting too long. So for now, at least, you can click on this picture to get an explanation of that!

### References

Besides what I’ve already mentioned about the classification of simple superalgebras, here are some other links. Wall proved a kind of super-Wedderburn theorem starting in the section of his paper called Elementary properties. Natalia Zhukavets has an Introduction to superalgebras which in Theorem 1.5 proves that in an algebraically closed field of characteristic different than 2, any simple superalgebra is of the form $<semantics>k[p|q]<annotation\; encoding="application/x-tex">k[p|q]</annotation></semantics>$ or $<semantics>D[n]<annotation\; encoding="application/x-tex">D[n]</annotation></semantics>$ where $<semantics>D=k[u]<annotation\; encoding="application/x-tex">D\; =\; k[u]</annotation></semantics>$, $<semantics>u<annotation\; encoding="application/x-tex">u</annotation></semantics>$ being an odd square root of 1. Over an algebraically closed field, this superdivision algebra $<semantics>D<annotation\; encoding="application/x-tex">D</annotation></semantics>$ is isomorphic to the division algebra that I called $<semantics>k[\sqrt{-1}]<annotation\; encoding="application/x-tex">k[\backslash sqrt\{-1\}]</annotation></semantics>$. Over a field that is not algebraically closed, they can be different, and there can be many nonisomorphic division algebras obtained by adjoining to $<semantics>k<annotation\; encoding="application/x-tex">k</annotation></semantics>$ an odd square root of $<semantics>a\in k<annotation\; encoding="application/x-tex">a\; \backslash in\; k</annotation></semantics>$ where $<semantics>a\ne 0<annotation\; encoding="application/x-tex">a\; \backslash ne\; 0</annotation></semantics>$.

Jinkui Wan and Weiqiang Wang have a paper with a Digression on superalgebras which summarizes Wall’s results in more modern language. Benjamin Gammage has an expository paper with a Classification of finite-dimensional simple superalgebras. This only classifies the ‘central’ ones — but as we’ve seen, that’s the key case.