The n-Category Café

Right now I’m more interested in the algebraic and geometrical aspects than their homotopy-theoretic consequences. So let’s think about Clifford algebras.

Let $Cliff_n$ be the free algebra over $\mathbb{R}$ generated by $n$ anticommuting square roots of $-1$. In its simplest algebraic form, Bott periodicity says that $Cliff_{n+8}$ is isomorphic to the algebra of $16 \times 16$ matrices with entries in $Cliff_n$:

$$Cliff_{n+8} \cong M_{16}(Cliff_n)$$

The only way I know to show this involves figuring out all the Clifford algebras. Luckily the first 8 are really interesting — I’ll talk about them later.

I’m interested in representations of Clifford algebras, so let $Rep(Cliff_n)$ be the category of real representations of $Cliff_n$. We’ll see what these are like in a minute: they play a fundamental role in representation theory. But there are really just 8 of them!

The reason is that when one algebra consists of all $k \times k$ matrices with entries in another algebra, their categories of representations are equivalent. Because they have equivalent categories of representation, we say these algebras are Morita equivalent.

So, $Cliff_{n+8}$ is Morita equivalent to $Cliff_n$, and we have an equivalence of categories

$$Rep(Cliff_{n+8}) \simeq Rep(Cliff_n)$$

For each $n$, we have an inclusion

$$Cliff_n \hookrightarrow Cliff_{n+1}$$

sending the generators of $Cliff_n$ to the first $n$ of the generators of $Cliff_{n+1}$. This lets us restrict any representation of $Cliff_{n+1}$ to a representation of $Cliff_n$, giving a functor

$$Rep(Cliff_{n+1}) \to Rep(Cliff_n)$$

I want to describe the Clifford algebras, their representation categories and these functors in detail. I would like to draw a large clock with 8 hours, one for each of the 8 Morita equivalence classes of Clifford algebras. I would like to write descriptions of the categories $Rep(Cliff_0), \dots , Rep(Cliff_8)$ on these 8 hours, and descriptions of the functors $Rep(Cliff_{n+1}) \to Rep(Cliff_n)$ labeling arrows going counterclockwise between these categories. This would be a nice version of the so-called ‘Bott clock’. But I don’t have the patience to do this right now. So I’ll adopt a more linear approach:

• $Cliff_0 \cong \mathbb{R}$. $Rep(Cliff_0)$ is the category of real vector spaces.

• $Cliff_1 \cong \mathbb{C}$. $Rep(Cliff_1)$ is the category of complex vector spaces. The functor $Rep(Cliff_1) \to Rep(Cliff_0)$ sends each complex vector space to its underlying real vector space.

• $Cliff_2 \cong \mathbb{H}$. $Rep(Cliff_2)$ is the category of ‘quaternionic vector spaces’ — that is, left $\mathbb{H}$-modules. The functor $Rep(Cliff_1) \to Rep(Cliff_0)$ sends each quaternionic vector space to its underlying complex vector space.

So far this is a nice simple progression, like the green shoots of grass growing in the spring. You might naively expect it to keep on going forever with octonions and hexadecanions and so on, just like grass keeps growing forever taller into the sky… but no, it doesn’t. Things change at this point:

• $Cliff_3 \cong \mathbb{H} \oplus \mathbb{H}$. $Rep(Cliff_3)$ is the category of $\mathbb{Z}_2$-graded quaternionic vector spaces — that is, quaternionic vector spaces $V$ that are split as a direct sum of two summands $V_0 \oplus V_1$. The inclusion $Cliff_2 \hookrightarrow Cliff_3$ maps any quaternion $q$ to the pair $(q,q)$. Thus, the functor $Rep(Cliff_3) \to Rep(Cliff_2)$ takes any $\mathbb{Z}_2$-graded quaternionic vector space and forgets the grading.

• $Cliff_4 \cong M_2(\mathbb{H})$. $Rep(Cliff_4)$ is equivalent to the category of quaternionic vector spaces again, since $M_2(\mathbb{H})$ is Morita equivalent to $\mathbb{H}$. The inclusion $Cliff_2 \hookrightarrow Cliff_3$ maps any pair of quaternions $(q,q')$ to the diagonal matrix $$\left( \begin{array}{cc} q & 0 \\ 0 & q' \end{array} \right)$$ so the functor $Rep(Cliff_4) \to Rep(Cliff_3)$ sends any quaternionic vector space $V$ to the $\mathbb{Z}/2$-graded quaternionic vector space $V \oplus V$. I will call this functor ‘doubling’.

The last two functors are adjoints of each other! I’ll say more about these adjoints later, but we no longer feel like we’re going ‘forward’ in an unambiguous sense: it feels like we’ve turned back. Indeed:

• $Cliff_5 \cong M_4(\mathbb{C})$. $Rep(Cliff_5)$ is equivalent to the category of complex vector spaces, again by Morita equivalence. What about the inclusion $Cliff_4 \hookrightarrow Cliff_5$? You can think of quaternions as special $2 \times 2$ matrices of complex numbers. You can do it in various way, but the theory of Clifford algebras picks out a specific one, $\mathbb{H} \hookrightarrow M_2(\mathbb{C})$. This in turn gives the inclusion we want, $M_2(\mathbb{H}) \hookrightarrow M_4(\mathbb{C})$. What about the functor $Rep(Cliff_5) \to Rep(Cliff_4)$? It’s really a functor from complex vector spaces to quaternionic vector spaces. It sends any complex vector space $V$ to $\mathbb{C}^2 \otimes_{\mathbb{C}} V$, which becomes a quaternionic vector space using the inclusion $\mathbb{H} \hookrightarrow M_2(\mathbb{C})$.

• $Cliff_6 \cong M_8(\mathbb{R})$. $Rep(Cliff_6)$ is equivalent to the category of real vector spaces, again by Morita equivalence. What about the inclusion $Cliff_5 \hookrightarrow Cliff_6$? You can think of complex numbers as special $2 \times 2$ matrices of real numbers. You can do it in various way, but the theory of Clifford algebras picks out a specific one, $\mathbb{C} \hookrightarrow M_2(\mathbb{R})$. This in turn gives the inclusion we want, $M_4(\mathbb{C}) \hookrightarrow M_8(\mathbb{R})$. What about the functor $Rep(Cliff_6) \to Rep(Cliff_5)$? It’s really a functor from real vector spaces to complex vector spaces. It sends any real vector space $V$ to $\mathbb{R}^2 \otimes_{\mathbb{R}} V$, which becomes a complex vector space using the inclusion $\mathbb{C} \hookrightarrow M_2(\mathbb{R})$.

As you can see, these two steps have a very similar flavor! The second functor is called ‘complexification’ so the first should be called ‘quaternionification’. In fact, these functors are adjoint to the very first two on our list. So we are now going backwards.

But despite having arrived back at the category of real vector spaces, we are not quite done!

• $Cliff_7 \cong M_8(\mathbb{R}) \oplus M_8(\mathbb{R})$. $Rep(Cliff_7)$ is equivalent to the category of $\mathbb{Z}/2$ graded real vector spaces, because $M_8(\mathbb{R}) \oplus M_8(\mathbb{R})$ is Morita equivalent to $\mathbb{R} \oplus \mathbb{R}$. The inclusion $Cliff_6 \hookrightarrow Cliff_7$ maps any $8 \times 8$ real matrix $T$ to the pair $(T,T)$. Thus, the functor $Rep(Cliff_8) \to Rep(Cliff_7)$ takes a $\mathbb{Z}/2$-graded vector space and forgets the grading.

• $Cliff_8 \cong M_{16}(\mathbb{R})$. $Rep(Cliff_8)$ is equivalent to the category of real vector spaces, by Morita equivalence, and now we are really back where we started. The inclusion $Cliff_7 \hookrightarrow Cliff_8$ maps any pair of $8 \times 8$ real matrices $(T,T')$ to the $16 \times 16$ matrix $$\left( \begin{array}{cc} T & 0 \\ 0 & T' \end{array} \right)$$ so the functor $Rep(Cliff_8) \to Rep(Cliff_7)$ sends any real vector space $V$ to the $\mathbb{Z}/2$-graded real vector space $V \oplus V$. This is again a form of ‘doubling’.

You’ll notice that this last pair of functors is suspiciously similar to the second pair we saw.

Now, all 8 functors we’ve seen have adjoints, which are other functors on our list.

“Left or right adjoints?” the category theorists wearily inquire, tired of people failing to say which. Both — in fact these functors all have ambidextrous adjoints, which are both left and right adjoints. I think this has something to do with the fact that Clifford algebras are semisimple. I could probably figure it out — this is not one of questions I meant to ask — but since you’re probably waiting for the questions to come along, I might as well ask:

Question 1. Suppose $A$ and $B$ are semisimple algebras over some field $k$ and $f: A \to B$ is a homomorphism. Does the ‘restriction of scalars’ functor $Rep(B) \to Rep(A)$ always have an ambidextrous adjoint?

So, I could summarize the story so far by drawing a clock with 8 hours, functors going clockwise from each hour to the next, and their ambidextrous adjoints going counterclockwise. But I will lazily draw this picture linearly, with hour 8 = hour 0 appearing both on top and on bottom. The functors I’ve already listed will point upward, and their adjoints will point down.

$$Rep(Cliff_0) \simeq [\text{real vector spaces}]$$

$$complexification \downarrow \uparrow \text{forgetting complex structure}$$

$$Rep(Cliff_1) \simeq [\text{complex vector spaces}]$$

$$quaternionification \downarrow \uparrow \text{forgetting quaternionic structure}$$

$$Rep(Cliff_2) \simeq [\text{quaternionic vector spaces}]$$

$$doubling \downarrow \uparrow \text{forgetting grading}$$

$$Rep(Cliff_3) \simeq [\mathbb{Z}/2\text{-graded quaternionic vector spaces}]$$

$$\text{forgetting grading} \downarrow \uparrow \text{doubling}$$

$$Rep(Cliff_4) \simeq [\text{quaternionic vector spaces}]$$

$$\text{forgetting quaternionic structure} \downarrow \uparrow \text{quaternionification}$$

$$Rep(Cliff_5) \simeq [\text{complex vector spaces}]$$

$$\text{forgetting complex structure} \downarrow \uparrow \text{complexification}$$

$$Rep(Cliff_6) \simeq [\text{real vector spaces}]$$

$$\text{doubling} \downarrow \uparrow \text{forgetting grading}$$

$$Rep(Cliff_7) \simeq [\mathbb{Z}/2\text{-graded real vector spaces}]$$

$$\text{forgetting grading} \downarrow \uparrow \text{doubling}$$

$$Rep(Cliff_8) \simeq [\text{real vector spaces}]$$

I’ll end with two small remarks.

If you’re paying close attention you may have noticed something funny: some of the categories and functors show up twice in this chart! The reason — or at least one reason — is that we’re treating the Clifford algebras as ordinary algebras, when in fact they are naturally $\mathbb{Z}/2$-graded. We should think of them as $\mathbb{Z}/2$-graded algebras generated by odd anticommuting square roots of $-1$. Two Clifford algebras that are Morita equivalent as ordinary algebras can be Morita inequivalent as $\mathbb{Z}/2$-graded algebras. This eliminates the redundancy we’re seeing here. For more on this, see:

• The Tenfold Way (Part 4): super division algebras and the Brauer–Wall groups of $\mathbb{R}$ and $\mathbb{C}$.

You may have also noticed a category that’s missing from the above chart: the category of $\mathbb{Z}/2$-graded complex vector spaces. We’d get that if we considered complex rather than real Bott periodicity. If $\mathbb{C}\!\, liff_n$ is the complex algebra generated by $n$ anticommuting square roots of $-1$, then

$$\mathbb{C}\!\, liff_0 = \mathbb{C}, \qquad \mathbb{C}\!\, liff_1 = \mathbb{C} \oplus \mathbb{C}$$

and

$$\mathbb{C}\!\, liff_{n+2} \cong M_2(\mathbb{C}\!\, liff_n)$$

so complex Bott periodicity has period 2. The complex analogue of our big chart looks like this:

$$Rep(\mathbb{C}\!\, liff_0) \simeq [\text{complex vector spaces}]$$

$$\text{doubling} \downarrow \uparrow \text{forgetting grading}$$

$$Rep(\mathbb{C}\!\, liff_1) \simeq [\mathbb{Z}/2\text{-graded complex vector spaces}]$$

$$\text{forgetting grading} \downarrow \uparrow \text{doubling}$$

$$Rep(\mathbb{C}\!\, liff_2) \simeq [\text{complex vector spaces}]$$

Again, the category at the bottom is a repeat of the category at top, due to Bott periodicity. It’s not so necessary this time.