Division Algebras, Clifford Algebras, Periodicity

Geoffrey Dixon

gdixon@7stones.com

The dimensions 2, 8 and 24 play signiﬁcant roles in lattice theory. In

Clifford algebra theory there are well-known periodicities of the ﬁrst two

of these dimensions. Using novel representations of the purely Euclidean

Clifford algebras over all four of the division algebras, R,C,H, and O, a

door is opened to a Clifford algebra periodicity of order 24 as well.

Introduction: Bott, Clifford Algebras, Lattices, and Notation

There are well-known periodicities in Clifford algebra (CA) theory of orders 2, 4, and

8 (see [1] for an introduction to Bott periodicity in the CA context, and [2]). These pe-

riodicities go hand-in-hand with matrix representations of CAs over the R(real num-

bers), C(complex numbers), and H(quaternion algebra). In most discussions of CA

representations the last division algebra in this sequence (the octonions, O), is left out.

In lattice theory the remarkable 24-dimensional Leech lattice ([6]) can be nicely

represented in (O,O,O), the 3-dimensional space with octonion components, so 24-

dimensional over R(see [3], [4], [5]).

Our goal here is to demonstrate that by exploiting the octonion algebra, O, in CA

representation theory a periodicity of order 24 arises, providing yet another link of the

algebra Oto the dimension 24.

My introduction to the mathematics of both CAs and division algebras - speciﬁcally

the real numbers R, complex numbers C, quaternions H, and octonions O- is [2]. Note:

notations have evolved since then, and CL(p, q )will denote the CA of a p,q-pseudo-

orthogonal space with metric signature, p(+), q(-).

I use the following matrix notations:

K(n)

the algebra of n×nmatrices over the division algebra K.

2K(n)

the block diagonal 2n×2nmatrices over K(n) : (so 2n2-dimensional). So, for exam-

ple, elements of K(2n)take the form

K(n)K(n)

K(n)K(n),

and elements of 2K(2n)take the form

K(n) 0

0K(n).

1

In particular, given this basis for R(2),

=1 0

0 1 , α =1 0

0−1, β =0 1

1 0 , γ =0 1

−1 0 ,

we have this as a basis for 2R,

=1 0

0 1 , α =1 0

0−1.

(All matrices will be dispensed with shortly.)

Further, for any algebra K, let

KLand KRand KA

denote the algebras of all actions of Kon itself from the left, the right, and both sides,

respectively. In the case of the octonions this requires nested actions due to nonasso-

ciativity (see [7] and [5]).

I shall restrict my focus here to the sequences of CAs, CL(k , 0) and CL(0, k). Since

CL(p+ 1, q + 1) ' CL(p, q)⊗R(2),

nothing is lost by this restriction of focus (and what I intend to do only works on these

ends).

Consider the following tables of CA isomorphisms (derived from [2]):

kCL(0, k)CL(k , 0)

0R

1C2R

2H R(2)

32H C(2)

4H(2)

5C(4) 2H(2)

6R(8) H(4)

72R(8) C(8)

8R(16)

Just to clarify,

CL(4,0) ' CL(0,4) 'H(2),

so I collapse those two isomorphisms to the center of the table.

Of particular importance,

CL(8,0) ' CL(0,8) 'R(16) ' CL(0,0) ⊗R(16).

This is the ﬁrst example of Bott periodicity of order 8 in the CA context. In general,

CL(k+ 8,0) ' CL(k, 0) ⊗R(16),

CL(0, k + 8) ' CL(0, k)⊗R(16).

2

Bott without Matrices

However, we can dispense with all matrix algebras by making use of split versions of

the division algebras. Bases for C,Hand Oare

C:{1, i};

H:{q0= 1, q1, q2, q3};

O:{e0:= 1, e1, e2, e3, e4, e5, e6, e7}

His noncommutative, but associative, and its multiplication table invariably begins

with (and is determined by),

q1q2=−q2q1=q3.

The multiplication table for Ois determined by specifying bases for 7 quaternionic sub-

algebras. Speciﬁcally, the most elegant of these has quaternionic triples given schemat-

ically by the 7 triples,

{e1+k, e2+k, e4+k},

k = 0 to 6, subscripts modulo 7, from 1 to 7. So, set k = 5, yielding the quaternionic

triple:

e6e7=−e7e6=e2.

(See [5] for multiplication tables and much more).

We now need a new copy of the complex algebra, and we’ll denote its imaginary

unit ι(so ι2=−1, and ιcommutes with everything, but it is not the same as our

original complex unit i). Then bases for split versions of those division algebras (using

the multiplication tables above) are

˜

C:{1, ιi};

˜

H:{q0= 1, q1, ιq2, ιq3};

˜

O:{e0:= 1, e1, e2, ιe3, e4, ιe5, ιe6, ιe7}

(although these are in fact real algebras, they are no longer division algebras; also,

just to be clear, this split version of the octonion algebra requires {e1, e2, e4}to be a

quaternionic triple, so it should be clear that these bases are not unique in the quaternion

and octonion cases; this is not important).

We rid ourselves of all matrix algebras by making use of the following isomor-

phisms and equivalencies:

˜

C'2R

H'HL'HR

˜

H'˜

HL'˜

HR'R(2)

˜

H2'H2'HA'R(4)

OL=OR=OA=˜

OL=˜

OR=˜

OA'R(8)

In this, and in what follows, it is understood that Kn:= K⊗K⊗... ⊗K, where there

are ndistinct copies of Kon the righthand side (see [7] and [5]).

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With these isomorphisms in hand I want to replace the Porteous table of CA isomor-

phisms above by rewriting it more schematically, using some different isomorphisms,

and without matrices:

Clifford algebra isomorphisms.

CL(0, k)kCL(k , 0)

R0R

C R 1R˜

C

HLR2R˜

HL

˜

C HLR3R˜

HLC

˜

HLHLR4R˜

HLHL

C˜

HLHLR5R˜

HLHL˜

C

HL˜

HLHLR6R˜

HLHL˜

HL

˜

C HL˜

HLHLR7R˜

HLHL˜

HLC

H4

L8H4

L

Complete by putting ⊗between algebras in CL(0, k)and CL(k , 0) columns.

We read from this table, for example, that

CL(0,6) 'HL⊗˜

HL⊗HL⊗R,

CL(6,0) '˜

HL⊗HL⊗˜

HL⊗R.

In the second line above the pieces of CL(6,0) are presented in reverse order to high-

light the major feature of this table: CL(k, 0) and CL(0, k)are ”split duals” when rep-

resented like this. That is, to get CL(k, 0) from C L(0, k)(or C L(0, k)from CL(k, 0)),

replace all its split parts by not split versions (so ˜

HL−→ HL), and replace all not split

versions with their split counterparts (so HL−→ ˜

HL).

Of course, these representations are not unique. For example, using the octonion

algebra we get

CL(0,6) 'OL.

Interestingly, ˜

OL'OL,

so the octonions cannot be exploited in this split duality picture as simply as Cand H,

but we shall see that they do have a part to play.

4

There are some striking periodicities in the table above. Modulo 2 we see that going

from k= 2nto k= 2n+ 1 we alternately add Cor ˜

C, which depending on if we are

looking at the CL(0, k)column, or CL(k , 0). And modulo 4 we see that

CL(0,4n)' CL(4n, 0), n ≥0.

Modulo 8 is the big periodicity, related to what is known as Bott periodicity. In this

context we ﬁrst see that at k= 8 there is a kind of algebraic collapse, or simpliﬁcation,

in the representation. But also,

CL(0, k + 8) ' CL(0, k)⊗ CL(0,8),

CL(k+ 8,0) ' CL(k, 0) ⊗ CL(0,8).

This kind of order 8 periodicity applies as well to CL(p, q ), with neither pnor qequal

to 0, but I’m not interested in that here. However, in that case we lose the split duality.

For example, CL(1,1) '˜

HL, which is not self-dual.

Lattices and Dimensions 1, 2, 8 and 24

I accumulated most of my ideas (and what I know) about lattice theory and sphere

packings in [5]. My interest in the Leech lattice, speciﬁcally, derives from [6], and it

relates to my investigations into the roles exceptional mathematical objects, like the

division algebras, play in theoretical physics ([7], [5], [10]). Conway and Sloane make

it abundantly apparent that the Leech lattice satisﬁes a great many criteria for excep-

tionality in this notoriously complex ﬁeld.

One of the leaders in the ﬁeld is Henry Cohn who wrote a paper summarizing a

recent breakthrough [8]. I’d like to share a few quotes. The initial breakthrough, the

work of M. S. Viazovska, related to the sphere packing problem in 8 dimensions [9].

She proved that E8(8-dimensional laminated lattice, also denoted Λ8) is the densest

sphere packing in 8 dimensions. Cohn says:

No proof of optimality had been known for any dimension above three, and

Viazovskas paper does not even address four through seven dimensions.

Cohn and collaborators then applied Viazovskas method to prove the Leech lattice

(Λ24) is the desnsest packing in 24 dimensions. And again, their work skirts all the

intermediate dimensions, 9 to 23. Cohn says:

Unfortunately, our low-dimensional experience is poor preparation for un-

derstanding high-dimensional sphere packing. Based on the ﬁrst three di-

mensions, it appears that guessing the optimal packing is easy, but this

expectation turns out to be completely false in high dimensions.

...

The sphere packing problem seems to have no simple, systematic solution

that works across all dimensions. Instead, each dimension has its own id-

iosyncracies and charm. Understanding the densest sphere packing in R8

tells us only a little about R7or R9, and hardly anything about R10.

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Aside from R8and R24, our ignorance grows as the dimension increases.

In high dimensions, we have absolutely no idea how the densest sphere

packings behave. We do not know even the most basic facts, such as

whether the densest packings should be crystalline or disordered. Here

”do not know” does not merely mean ”cannot prove,” but rather the much

stronger ”cannot predict.”

What’s going on here? Why are dimensions 8 and 24 so amenable to proof, and no

other high dimensional lattice (none; not one)? The laminated lattices in dimensions

1 and 2 are nice, but the hellish complexity so common in lattice theory begins in

dimension 3, and only disappears in dimensions 8 and 24 thereafter.

There are four division algebras associated with parallelizable spheres. These occur

in dimensions

1,2,4,8.

And now we have a new ﬁnite sequence of exceptional dimensions revolving around

lattice theory:

1,2,8,24.

Taking these four numbers and dividing by the previous 4, we get

1,1,2,3,

the beginning of the Fibonacci sequence (I mentioned this stuff in [5]). (This could

be mere coincidence, what is in contemporary mathematical parlance referred to as

moonshine ([11]).)

One more word about the dimensions 1, 2, 8 and 24. Cohn and Elkies ([8]) devel-

oped upper bounds (linear programming bounds) for sphere packings in kdimensions.

These bounds vary smoothly, unlike the actual densities of sphere packings that tend to

bounce about in a distinctly discontinuous manner. There are four dimensions where

the maximal known lattice density in any dimension achieves this upper bound (or

appears to to several signiﬁcant ﬁgures): 1, 2, 8 and 24.

Split Dual Clifford Algebra Table up to k= 24

Let’s take a look at the split dual CA table introduced above, but now expanded to

k= 24:

6

Clifford algebra isomorphisms to dimension 24.

CL(0, k)kCL(k , 0)

R0R

C R 1R˜

C

HLR2R˜

HL

˜

C HLR3R˜

HLC

˜

HLHLR4R˜

HLHL

C˜

HLHLR5R˜

HLHL˜

C

HL˜

HLHLR6R˜

HLHL˜

HL

˜

C HL˜

HLHLR7R˜

HLHL˜

HLC

H4

L8H4

L

C H4

L9H4

L˜

C

HLH4

L10 H4

L˜

HL

˜

C HLH4

L11 H4

L˜

HLC

˜

HLHLH4

L12 H4

L˜

HLHL

C˜

HLHLH4

L13 H4

L˜

HLHL˜

C

HL˜

HLHLH4

L14 H4

L˜

HLHL˜

HL

˜

C HL˜

HLHLH4

L15 H4

L˜

HLHL˜

HLC

H8

L16 H8

L

C H8

L17 H8

L˜

C

HLH8

L18 H8

L˜

HL

˜

C HLH8

L19 H8

L˜

HLC

˜

HLHLH8

L20 H8

L˜

HLHL

C˜

HLHLH8

L21 H8

L˜

HLHL˜

C

HL˜

HLHLH8

L22 H8

L˜

HLHL˜

HL

˜

C HL˜

HLHLH8

L23 H8

L˜

HLHL˜

HLC

O4

L24 O4

L

Complete by putting ⊗between algebras in CL(0, k)and CL(k , 0) columns.

This table makes the order 8 periodicity very pronounced. At every multiple of 8

there is a kind of algebraic collapse/simpliﬁcation, after which we start adding things

in the same way as we did previously. Keep in mind that few of these representations

are unique. For example, at k = 16,

H4

A'H8

L'HA⊗O2

L.

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So the octonion algebra could have been introduced before k = 24.

However, 24 is the ﬁrst dimension for which CL(0, k)' C L(k, 0), and both can be

represented by tensored copies of OL(necessarily the same). There is a theme running

through this mathematical realm that has arisen elsewhere (see, for example, [5]):

H4

L' CL(8,0) ' CL(0,8),

O4

L' CL(24,0) ' CL(0,24).

That is, the quaternions are associated with dimension 8, and the octonions with di-

mension 24.

[Word of explication: 24 is the smallest dimension k for which

CL(k, 0) ' CL(0, k ),

and both can be represented purely in terms of OL(O4

L). The ﬁrst dimension in which

any Clifford algebra can feature the full OLin its representation is n= 6. And the

ﬁrst dimension in which all CL(p, q )can exploit OLas part of their representations is

p+q= 8.

24 = LCM (6,8).

LCM is the least common multiple.]

Let’s take a look at a 1-vector basis for the Clifford algebra CL(24,0) represented

by O4

L. We need four copies of O, and we’ll denote their bases by

mea, a = 0, ..., 7, m = 1,2,3,4.

This is the CL(24,0) 1-vector basis I came up with (p= 1, ..., 6; multiply across rows):

1eLp 2eL73eL04eL0

1eL02eLp 3eL74eL0

1eL72eL03eLp 4eL0

1eL72eL73eL74eLp

This gives us 24 anti-commuting elements of O4

L(6 for each row). The product of all

24 is

±1eL7

2eL7

3eL7

4eL7.

Interestingly, if we replace Oby H(that is, H4

L), and build a similar basis for a

Clifford algebra using quaternions instead of octonions (r= 1,2below), we get

1qLr 2qL33qL04qL0

1qL02qLr 3qL34qL0

1qL32qL03qLr 4qL0

1qL32qL33qL34qLr

which is a basis for CL(8,0). So the octonions are associated with CL(24,0), and the

quaternions with CL(8,0), at least within this context.

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Lattices, Clifford Algebras, Periodicity

The question is: is this order 24 algebraic collapse to a product of just octonions (left

actions) meaningful? It recurs at every dimension 24m,ma positive integer, so it is

periodic.

Topologically Bott periodicity has to do with homotopy groups and the sequences

of classical Lie groups, orthogonal, unitary and symplectic. In this context the primary

kinds of periodicities that arise are of order 2, 4 and 8.

In the theory of laminated lattices there are also indications of periodicities of order

2, 4, 8 and 24 (see [5]). It was this that inspired this look at Clifford algebra period-

icity, and in particular the tantalizing representational collapse O4

L' CL(24,0) '

CL(0,24). At dimension 24myou get the collapse

O4m

L' CL(24m, 0) ' CL(0,24m)

so at least in the Clifford algebra context there is an algebraic periodicity of order 24,

as well as 8 (which is another manifestation of Bott periodicity).

The question naturally arises: is there a topological periodicity of order 24 asso-

ciated with this algebraic periodicity (as there is of order 8)? Can this question even

be answered given our present mathematical machinery? Is this more than just moon-

shine? I suggest it is much more.

References

[1] K. Siegel,

http://math.stanford.edu/ ksiegel/BottPeriodicityAndCliffordAlgebras.pdf.

[2] I.R. Porteous, Topological Geometry, (Cambridge, 2nd Ed., 1981).

[3] G.M. Dixon, Octonions: Invariant Leech Lattice Exposed, hep-th 9506080.

[4] R.A. Wilson, Octonions and the Leech Lattice, J. of Algebra, Vol 322, Issue 6

(2009), 2186-2190.

[5] G.M. Dixon, Division Algebras, Lattices, Physics, Windmill Tilting, (CreateSpace,

2011).

[6] J.H. Conway, N.J.A. Sloane, Sphere Packings, Lattices and Groups, (Springer-

Verlag, 2nd Ed., 1991).

[7] G.M. Dixon, Division Algebras: Octonions, Quaternions, Complex Numbers and

the Algebraic Design of Physics, (Kluwer (now Springer-Verlag), 1994).

[8] Henry Cohn, A Conceptual Breakthrough in Sphere Packing, (Notices of the Amer-

ican Mathematical Society 64(02) November 2016).

9

[9] M. S. Viazovska, The sphere packing problem in dimension 8, (preprint, 2016.

arXiv:1603.04246).

[10] G.M. Dixon, Seeable Matter; Unseeable Antimatter, Comment. Math. Univ. Car-

olin. 55,3 (2014) 381-386.

[11] E. Klarreich, Mathematicians Chase Moonshines Shadow,

https://www.quantamagazine.org/mathematicians-chase-moonshine-string-

theory-connections-20150312/.

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