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composition algebra

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Definition

Let kk be a field with characteristic char(k)2char(k) \neq 2. A composition algebra over kk consists of a finite-dimensional vector space VV together with a

  • A nondegenerate symmetric bilinear form ,:VVk\langle , \rangle: V \otimes V \to k,

  • A multiplication map, i.e., a linear map :VVV- \cdot -: V \otimes V \to V,

  • A unit eVe \in V for the multiplication, i.e., so that ev=v=vee \cdot v = v = v \cdot e,

such that, putting N(v)=v,vN(v) = \langle v, v \rangle,

  • N(u)N(v)=N(uv)N(u) N(v) = N(u v) (writing uvu v for uvu \cdot v).

There are no assumptions on the multiplication such as associativity, commutativity, etc. Examples of composition algebras include the real numbers, the complex numbers, the quaternions, the octonions, and the algebra of 2×22 \times 2 matrices over a field.

Since char(k)2char(k) \neq 2, we can recover the bilinear form from the norm by the formula

u,v=N(u+v)N(u)N(v)2\langle u, v \rangle = \frac{N(u + v) - N(u) - N(v)}{2}

Since the bilinear form is nondegenerate, we may infer u=vu = v whenever

u,w=v,w\langle u, w \rangle = \langle v, w \rangle

and this will be frequently used in the sequel.

Also since the form is nondegenerate, there exists vVv \in V such that N(v)0N(v) \neq 0. From N(v)=N(ev)=N(e)N(v)N(v) = N(e v) = N(e)N(v), it follows that N(e)=1N(e) = 1.

Properties

Basic identities

The arrangements of the proofs below are based in part on the treatments by Conway and Smith, and by Springer and Veldkamp (see references below).

Proposition (Scaling)

uv,uw=N(u)v,w\langle u v, u w \rangle = N(u)\langle v, w \rangle and uw,vw=u,vN(w)\langle u w, v w \rangle = \langle u, v \rangle N(w)

Proof

The left sides, and therefore the right sides of the equations below are equal:

N(u(v+w))=N(u)N(v+w)=N(u)(N(v)+2v,w+N(w))N(u(v + w)) = N(u)N(v + w) = N(u)(N(v) + 2\langle v, w \rangle + N(w))

[]

N(uv+uw)=N(uv)+2uv,uw+N(uw)=N(u)N(v)+2uv,uw+N(u)N(w)N(u v + u w) = N(u v) + 2\langle u v, u w \rangle + N(u w) = N(u)N(v) + 2\langle u v, u w \rangle + N(u)N(w)

and the result follows by cancellation and division by 22.

Proposition (Exchange)

uv,wx=2u,wv,xux,wv\langle u v, w x \rangle = 2\langle u, w \rangle \langle v, x \rangle - \langle u x, w v \rangle

Proof

From the scaling identity, we have

(u+w)v,(u+w)x=N(u+w)v,x=N(u)v,x+2u,wv,x+N(w)v,x=uv,ux+2u,wv,x+wv,wx\langle (u + w)v, (u + w)x \rangle = N(u + w)\langle v, x \rangle = N(u)\langle v, x \rangle + 2\langle u, w \rangle \langle v, x \rangle + N(w)\langle v, x \rangle = \langle u v, u x \rangle + 2\langle u, w \rangle \langle v, x \rangle + \langle w v, w x \rangle

but the left-hand side is equal to

uv+wv,ux+wx=uv,ux+wv,ux+uv,wx+wv,wx\langle u v + w v, u x + w x \rangle = \langle u v, u x \rangle + \langle w v, u x \rangle + \langle u v, w x \rangle + \langle w v, w x \rangle

and now we equate the right-hand sides and cancel to get the result.

Conjugation identities

In any composition algebra, we may define a conjugation operator by

v¯=2v,eev\bar{v} = 2\langle v, e \rangle e - v

Observe that v¯=v\bar{v} = v just when vv is a scalar multiple of the identity. By analogy with the classical case (composition algebras over \mathbb{R}), such elements will be called real.

The next few propositions develop properties of conjugation.

Proposition (Adjointness)

uv,w=v,u¯w\langle u v, w \rangle = \langle v, \bar{u}w \rangle and uv,w=u,wv¯\langle u v, w \rangle = \langle u, w\bar{v} \rangle. w,uv=u¯w,v\langle w, u v \rangle = \langle \bar{u} w, v \rangle and w,uv=wv¯.u\langle w, u v \rangle = \langle w\bar{v}. u \rangle.

Proof

Put x=ex = e in the exchange identity to get the first equation in

uv,w=2u,wv,eu,wv=u,w(2v,eev)=u,wv¯\langle u v, w \rangle = 2\langle u, w \rangle \langle v, e \rangle - \langle u, w v \rangle = \langle u, w(2\langle v, e\rangle e - v)\rangle = \langle u, w\bar{v} \rangle

The second adjointness equation is proved similarly; the final two come from symmetry of the form.

Proposition (Involution)

v=v¯¯v = \bar{\bar{v}} for all vv.

Proof

For all uu we have

u,v=uv¯,e=u,v¯¯\langle u, v \rangle = \langle u\bar{v}, e \rangle = \langle u, \bar{\bar{v}} \rangle

and the result follows from nondegeneracy.

Proposition (Unitarity)

u,v=v¯,u¯=u¯,v¯\langle u, v \rangle = \langle \bar{v}, \bar{u} \rangle = \langle \bar{u}, \bar{v} \rangle.

Proof

u,v=e,u¯v=v¯,u¯=u¯,v¯\langle u, v \rangle = \langle e, \bar{u}v \rangle = \langle \bar{v}, \bar{u} \rangle = \langle \bar{u}, \bar{v} \rangle where the last equation is symmetry of the bilinear form.

Proposition (Anti-automorphism)

u¯v¯=vu¯\bar{u} \bar{v} = \widebar{v u}.

Proof

For all ww we have

u¯v¯,w=v¯,uw=v¯w¯,u=w¯,vu=vu¯,w\langle \bar{u}\bar{v}, w \rangle = \langle \bar{v}, u w \rangle = \langle \bar{v}\bar{w}, u \rangle = \langle \bar{w}, v u \rangle = \langle \widebar{v u}, w \rangle

using involution and unitarity. The result follows from nondegeneracy of the form.

By the involution and anti-automorphism properties, we see that v¯v\bar{v}v is fixed under conjugation: is “real”. Better yet,

Proposition (Reality)

u¯(uv)=N(u)v\bar{u} \cdot (u v) = N(u)v.

Proof

For all ww,

u¯(uv),w=uv,uw=N(u)v,w=N(u)v,w\langle \bar{u}\cdot (u v), w \rangle = \langle u v, u w \rangle = N(u)\langle v, w \rangle = \langle N(u)v, w \rangle

and the result follows from nondegeneracy.

This last result has several interesting corollaries. Putting v=ev = e, we see that

  • N(u)0N(u) \neq 0 implies uu is invertible, with u 1=u¯/N(u)u^{-1} = \bar{u}/N(u).

  • N(u)=0N(u) = 0 implies uu is a zero divisor, with u¯u=0\bar{u} u = 0.

In either case, we have from u¯=2u,eeu\bar{u} = 2\langle u, e \rangle e - u the identity

u¯u=(2u,eeu)u=N(u)e\bar{u} u = (2\langle u, e \rangle e - u)u = N(u)e

so that every element uu of a composition algebra satisfies a quadratic equation

u 22u,eu+N(u)e=0.u^2 - 2\langle u, e \rangle u + N(u)e = 0.

This has as further consequence the fact that an algebra admits at most one norm making it a composition algebra (because the minimal monic polynomial of an element uu in a finite-dimensional algebra is uniquely determined; the norm of an element would the uniquely determined constant coefficient of its minimal polynomial).

A final corollary of Reality is

Proposition (Alternative law)

u(uv)=u 2vu \cdot (u v) = u^2 \cdot v and uv 2=(uv)vu \cdot v^2 = (u v) \cdot v.

Proof

We have w(uv)=(wu)vw(u v) = (w u)v if ww is either ee or u¯\bar{u}, and uu is a linear combination of ee and u¯\bar{u}. The other equation is proven similarly.

These are the two axioms as given in alternative algebra, but we remark that often a third alternative law is considered: u(vu)=(uv)uu (v u) = (u v) u. For discussion of this in composition algebras, see the section on Moufang identities below.

Cayley-Dickson doubling construction

This is essentially the same as the Cayley-Dickson construction, but in this section it is applied specifically to composition algebras where we have to deal with a norm, whereas the general construction applies to general (nonassociative) algebras equipped with an anti-involution.

We begin with a simple observation:

Proposition

Let VV be a finite-dimensional vector space with a nondegenerate bilinear form, and let WW be a subspace such that the form on VV restricts to a nondegenerate form on WW. Then

V=WW V = W \oplus W^\perp

and the form on VV restricts to a nondegenerate form on W W^\perp.

Proof

The fact that WW ={0}W \cap W^\perp = \{0\} is immediate from nondegeneracy of the form on WW, and that W+W =VW + W^\perp = V follows from this and the fact that dim(W)+dim(W )=dim(V)dim(W) + dim(W^\perp) = dim(V) (use dim(W )=dim((V/W) *)=dim(V/W)dim(W^\perp) = dim((V/W)^*) = dim(V/W) and dim(V)=dim(W)+dim(V/W)dim(V) = dim(W) + dim(V/W)). For the second assertion, we know that for vW v \in W^\perp, the map v, W:Wk\langle v, - \rangle |_W: W \to k is zero; if also v, W :W k\langle v, - \rangle |_{W^\perp}: W^\perp \to k is zero, then v,:Vk\langle v, - \rangle: V \to k is zero because V=W+W V = W + W^\perp, and v=0v = 0 follows from nondegeneracy of the form on VV.

Thus, given a composition algebra VV and a composition subalgebra WW of VV (that is, a subspace closed under identity and multiplication, such that the norm on VV restricts to a nondegenerate form on WW), the proposition shows there exists αW \alpha \in W^\perp such that N(α)0N(\alpha) \neq 0. This α\alpha is invertible, so αW\alpha \cdot W has the same dimension as WW. Moreover, for all v,wWv, w \in W we have

αv,w=α,wv¯=0\langle \alpha v, w \rangle = \langle \alpha, w \bar{v} \rangle = 0

so that, by nondegeneracy of the form on WW, αWW={0}\alpha W \cap W = \{0\}. Indeed, αW\alpha W is orthogonal to WW. It follows that W+αWW + \alpha W has double the dimension of WW.

Now let us fix such an α\alpha, and put λ=N(α)\lambda = N(\alpha).

Proposition

For elements u,v,w,xWu, v, w, x \in W,

u+αv,w+αx=u,w+λv,x.\langle u + \alpha v, w + \alpha x \rangle = \langle u, w \rangle + \lambda \langle v, x \rangle.
Proof

This follows from the equations

u,αx=ux¯,α=0αv,w=α,wv¯=0αv,αx=N(α)v,x\langle u, \alpha x \rangle = \langle u \bar{x}, \alpha \rangle = 0 \qquad \langle \alpha v, w \rangle = \langle \alpha, w \bar{v} \rangle = 0 \qquad \langle \alpha v, \alpha x \rangle = N(\alpha)\langle v, x \rangle

plus bilinearity of the form.

Consequently, if u+αv,w=0\langle u + \alpha v, w \rangle = 0 for all wWw \in W, we must have u=0u = 0, and if u+αv,αx=0\langle u + \alpha v, \alpha x \rangle = 0 for all xWx \in W, then v=0v = 0. It follows that the form on VV, when restricted to W+αWW + \alpha W, is nondegenerate.

Now we want to show that the double W+αWW + \alpha W is closed under multiplication, hence forms a composition subalgebra. It follows immediately from all this that, starting from the trivial composition subalgebra kek \cdot e of dimension 1, dim(V)dim(V) must be a power of 2, and in fact we will see later that the only possible dimensions are 1, 2, 4, and 8. Indeed, the possible structures of composition algebras are very tightly constrained.

Proposition (Conjugation on the double)

We have u+αv¯=u¯αv\widebar{u + \alpha v} = \bar{u} - \alpha v. Consequently, αv=αv¯=v¯α¯=v¯α\alpha v = - \widebar{\alpha v} = - \bar{v} \bar{\alpha} = \bar{v} \alpha, and α¯=α\widebar{\alpha} = -\alpha.

Proof

αv¯=2αv,eeαv=αv\widebar{\alpha v} = 2\langle \alpha v, e \rangle e - \alpha v = -\alpha v.

Theorem (Closure under multiplication)

For all u,v,w,xWu, v, w, x \in W, (u+αv)(w+αx)=(uwλxv¯)+α(wv+u¯x)(u + \alpha v)(w + \alpha x) = (u w - \lambda x \bar{v}) + \alpha (w v + \bar{u} x).

Proof

For all yVy \in V, we have the following sets of equations, using the previous proposition (Conj), the Exchange identity (Ex), and other identities frequently observed above (unlabeled as such).

u(αx),y=αx,u¯y=Ex0αy,u¯x=y,α(u¯x)=α(u¯x),y\langle u (\alpha x), y \rangle = \langle \alpha x, \bar{u} y \rangle \stackrel{Ex}{=} 0 - \langle \alpha y, \bar{u} x \rangle = \langle y, \alpha (\bar{u} x) \rangle = \langle \alpha (\bar{u} x), y \rangle

[]

(αv)w,y=αv,yw¯=Conjv¯α,yw¯=Ex0v¯w¯,yα=(v¯w¯)α,y=wv¯α,y=Conjα(wv),y\langle (\alpha v)w, y \rangle = \langle \alpha v, y \bar{w} \rangle \stackrel{Conj}{=} \langle \bar{v} \alpha, y \bar{w} \rangle \stackrel{Ex}{=} 0 - \langle \bar{v}\bar{w}, y \alpha \rangle = \langle (\bar{v}\bar{w})\alpha, y \rangle = \langle \widebar{w v} \alpha, y \rangle \stackrel{Conj}{=} \langle \alpha (w v), y \rangle

[]

(αv)(αx),y=Conjαv,y(αx)=Ex0+α(αx),yv=Conjαx,α(yv)=λx,yv=λxv¯,y\langle (\alpha v)(\alpha x), y \rangle \stackrel{Conj}{=} -\langle \alpha v, y (\alpha x) \rangle \stackrel{Ex}{=} 0 + \langle \alpha (\alpha x), y v \rangle \stackrel{Conj}{=} -\langle \alpha x, \alpha (y v) \rangle = -\lambda \langle x, y v \rangle = \langle -\lambda x\bar{v}, y \rangle

These identities, combined with nondegeneracy of the form, give the result.

Possible dimensions are 1, 2, 4, and 8.

The calculation expressed by the fundamental theorem just stated has some remarkable consequences:

  • Suppose V=W+αWV = W + \alpha W. Then WW is an associative composition algebra.

For, by starting from the identity

N(u+αv)N(w+αx)=N((uwλxv¯)+α(wv+u¯x))N(u + \alpha v)N(w + \alpha x) = N((u w - \lambda x \bar{v}) + \alpha (w v + \bar{u} x))

and expanding, one obtains

(N(u)+λN(v))(N(w)+λN(x))=N(uw)2λuw,xv¯+λ 2N(xv¯)+λ(N(wv)+2wv,u¯x+N(u¯x))(N(u) + \lambda N(v))(N(w) + \lambda N(x)) = N(u w) - 2\lambda \langle u w, x\bar{v} \rangle + \lambda^2 N(x\bar{v}) + \lambda (N(w v) + 2\langle w v, \bar{u} x \rangle + N(\bar{u}x))

Using the fact that NN is a homomorphism, plus unitarity N(u)=N(u¯)N(u) = N(\bar{u}), further expansions and cancellations yield

0=2λuw,xv¯+2λwv,u¯x0 = -2\lambda \langle u w, x\bar{v} \rangle + 2\lambda \langle w v, \bar{u}x \rangle

which, by adjointness, yields

(uw)v,x=u(wv),x\langle (u w)v, x \rangle = \langle u(w v), x \rangle

which by nondegeneracy on WW, yields associativity (uw)v=u(wv)(u w)v = u(w v).

  • Suppose V=W+αWV = W + \alpha W is an associative composition algebra. Then WW is a commutative, associative composition algebra.

For clearly the subalgebra WW must be associative; it is also commutative via the following string of equations (using conjugation of the double):

α(vw)=(αv)w=(v¯α)w=v¯(αw)=v¯(w¯α)=(v¯w¯)α=wv¯α=α(wv)\alpha(v w) = (\alpha v)w = (\bar{v}\alpha)w = \bar{v}(\alpha w) = \bar{v}(\bar{w} \alpha) = (\bar{v}\bar{w})\alpha = \widebar{w v}\alpha = \alpha (w v)

and cancelling out α\alpha.

Conversely, a lengthy but straightforward calculation shows that if WW is commutative and associative, then VV is associative.

  • Suppose V=W+αWV = W + \alpha W is a commutative associative composition algebra. Then WW is purely real, i.e., is the trivial 1-dimensional associative commutative algebra kek \cdot e.

This results from

αw=Conjw¯α=commαw¯\alpha w \stackrel{Conj}{=} \bar{w}\alpha \stackrel{comm}{=} \alpha \bar{w}

so that w=w¯w = \bar{w} for every wWw \in W, so that ww is real. Conversely, from

(u+αv)(w+αx)=(uwλxv¯)+α(wv+u¯x)(u + \alpha v)(w + \alpha x) = (u w - \lambda x\bar{v}) + \alpha (w v + \bar{u} x)

[]

(w+αx)(u+αv)=(wuλvx¯)+α(ux+w¯v)(w + \alpha x)(u + \alpha v) = (w u - \lambda v\bar{x}) + \alpha (u x + \bar{w} v)

together with commutativity and trivial conjugation in WW, we infer commutativity in VV.

Hence the doubling process may be iterated three times at most.

This same result can also be proven using string diagram calculus. See this paper for a nice exposition of that route.

Hurwitz’s Theorem

The classification of composition algebras over specific fields (e.g., number fields, local fields) can be a bit intricate; in this section we concentrate solely on the classical case where k=k = \mathbb{R}, where the results have been known for a long time.

A fundamental dichotomy is whether or not the composition algebra has zero divisors, i.e., elements vv such that N(v)=0N(v) = 0. If not, then the composition algebra is a division algebra (every nonzero element is invertible). If so, then the composition algebra is called a split composition algebra. We analyze each in turn.

Proposition

In a division composition algebra, all nonzero elements have positive norm.

Proof

If all elements vv orthogonal to the identity ee have positive norm, the result is immediate since

N(rv+se)=r 2N(v)+s 20N(r v + s e) = r^2 N(v) + s^2 \geq 0

Otherwise, if some such element vv has N(v)=λ<0N(v) = \lambda \lt 0, we may put u=v/λ 1/2u = v/|\lambda|^{1/2} so that N(u)=1N(u) = -1. Then uu is orthogonal to ee and

N(u+e)=N(u)+N(e)=1+1=0N(u + e) = N(u) + N(e) = -1 + 1 = 0

which contradicts the assumption that all nonzero elements are invertible.

In particular, any division composition algebra is a normed division algebra.

Now let VV be a division composition algebra, with V=W+αWV = W + \alpha W, where 0αW 0 \neq \alpha \in W^\perp. Put j=α/N(α) 1/2j = \alpha/N(\alpha)^{1/2}, so that N(j)=1N(j) = 1, jWj \perp W, and V=W+jWV = W + j W. We have the following possibilities.

  • dim(V)=2dim(V) = 2. In that case WW is purely real and VV is a commutative field over \mathbb{R} with j 2=jj¯=N(j)=1-j^2 = j\bar{j} = N(j) = 1. This is of course the complex numbers, with

    N(s+jt)=s 2+t 2N(s + j t) = s^2 + t^2

    the usual norm. The conjugate of s+jts + j t is sjts - j t.

  • dim(V)=4dim(V) = 4. In that case WW is a 2-dimensional division composition algebra, hence isomorphic to \mathbb{C}, and VV is an associative division algebra over \mathbb{R} given by V=+jV = \mathbb{C} + j\mathbb{C}, where again j 2=1j^2 = -1. (Evidently VV is not commutative because WW is not purely real.) By conjugation of the double, we have

    ji=ijj i = -i j

    where ii is an imaginary unit of \mathbb{C}, and we arrive at the algebra of quaternions \mathbb{H} over \mathbb{R}, with orthonormal basis provided by 1,i,j,k=ij1, i, j, k = i j. Conjugation is given by the usual operation

    a+bi+cj+dkabicjdka + b i + c j + d k \mapsto a - b i - c j - d k
  • dim(V)=8dim(V) = 8. In that case WW is a 4-dimensional division composition algebra, hence isomorphic to \mathbb{H}, and VV is an alternative division algebra over \mathbb{R} given by V=+jV = \mathbb{H} + j\mathbb{H}, with j 2=1j^2 = -1. (VV is not associative because WW is not commutative.) The structure of multiplication is given by the theorem above and the resulting algebra is the algebra of octonions, with the standard norm and conjugation.

Thus, we have established

Theorem (Hurwitz)

The only division composition algebras over \mathbb{R} are the reals, complexes, quaternions, and octonions.

Split composition algebras

Now we turn to split composition algebras VV. It turns out that the structure of these is not specific to the field \mathbb{R}: the classification of possible split composition algebras is the same over any field (see the text by Springer and Veldkamp), although we will continue to work over \mathbb{R} as we describe them below.

Suppose V=W+αWV = W + \alpha W, where αW \alpha \in W^\perp, N(α)0N(\alpha) \neq 0. Put j=α/N(α) 1/2j = \alpha/|N(\alpha)|^{1/2}, so N(j)=1|N(j)| = 1, V=W+jWV = W + j W. In addition to the trivial 1-dimensional case, we have the following possibilities.

  • dim(V)=2dim(V) = 2. In this case N(j)=1N(j) = -1 (else VV would be a division algebra, not a split composition algebra) and j 2=1j^2 = 1 (we are now using 11 to denote the identity). The elements

    e 1=1+j2e 2=1j2e_1 = \frac{1 + j}{2} \qquad e_2 = \frac{1-j}{2}

    are primitive idempotents, conjugate to one another, and Ve 1e 2V \cong \mathbb{R} e_1 \oplus \mathbb{R} e_2 as a product ring. The norm of an element xe 1+ye 2x e_1 + y e_2 is N(xe 1+ye 2)=xyN(x e_1 + y e_2) = x y.

  • dim(V)=4dim(V) = 4. Let ii be an imaginary unit of WW, so i¯=i\bar{i} = -i and N(i)=1|N(i)| = 1. Here either N(i)=1N(i) = -1 (WW is split), or N(i)=1N(i) = 1 (WW is isomorphic to \mathbb{C}). In the second instance, N(j)=1N(j) = -1, else VV would be a division algebra, and we may replace WW by the split algebra W=+ijW' = \mathbb{R} + \mathbb{R} i j and still have V=W+jWV = W' + j W'. So without loss of generality we may assume WW is split; therefore, there is up to isomorphism only one split composition algebra of dimension 4. This is the algebra of 2×22 \times 2 matrices AA, for which N(A)=det(A)N(A) = det(A) and WW is embedded as the subalgebra of diagonal matrices; the element jj may be taken to be the matrix AA with a 11=a 22=0a_{11} = a_{22} = 0, a 12=a 21=1a_{12} = a_{21} = 1. The conjugate of a matrix AA is A¯=Tr(A)IA\bar{A} = Tr(A)I - A, which leads to the familiar formula for det(A)A 1det(A) A^{-1} when AA is invertible.

  • dim(V)=8dim(V) = 8. Again, by an argument similar to the one used for the case of dimension 4, we may assume a maximal proper composition subalgebra WW is split, and up to isomorphism there is only one split composition algebra of dimension 8, aka the split octonions. The multiplication may be deduced from the fundamental result on doubling multiplication above, or may be expressed as follows. Denote scalars by letters like r,sr, s and 3-vectors by letters like x,yx, y. Let x,y\langle x, y \rangle denote the standard inner product

    x 1y 1+x 2y 2+x 3y 3x_1 y_1 + x_2 y_2 + x_3 y_3

    and let xyx \wedge y denote the standard cross-product, so that xy,z=det(x,y,z)\langle x \wedge y, z \rangle = det(x, y, z). Elements of VV are represented by 2×22 \times 2 arrays

    (r x y s)\left( \begin{aligned} r & x\\ y & s \end{aligned} \right)

    and multiplication is given by the following formula, highly reminiscent of matrix multiplication but with some cross-product cross terms:

    (r x y s)(r x y s)=(rr+x,y rx+sx+yy ry+sy+xx y,x+ss)\left( \begin{aligned} r & x\\ y & s \end{aligned} \right) \cdot \left( \begin{aligned} r' & x'\\ y' & s' \end{aligned} \right) = \left( \begin{aligned} r r' + \langle x, y' \rangle & r x' + s' x + y \wedge y'\\ r' y + s y' + x \wedge x' & \langle y, x' \rangle + s s' \end{aligned} \right)

    The norm is given by a kind of determinant formula

    N(r x y s)=rsx,yN\left( \begin{aligned} r & x\\ y & s \end{aligned} \right) = r s - \langle x, y \rangle

Moufang identities

Further consequences of the composition algebra axioms include the Moufang laws which are important in the study of octonions.

Moufang identities

  • (uv)(wu)=(u(vw))u)=u((vw)u)(u v)(w u) = (u(v w))u) = u((v w)u)

  • ((uv)u)w=u(v(uw))((u v)u)w = u(v(u w))

  • ((uv)w)v=u(v(wv))((u v)w)v = u(v(w v))

We will prove the first of these; the others are proven in similar style (see Springer-Veldkamp for details). (It may be tricky to remember how the bracketings go, but one thing to remember is that the bracketings shouldn’t lead to proofs of general associativity when interpreted in a division algebra!)

Proof

We have

(uv)(wu),x = uv,x(u¯w¯) =Ex 2u,xv,u¯w¯u(u¯w¯),xv = 2u,xvw,u¯u¯w¯,u¯(xv) = 2vw,u¯u,xN(u)w¯v¯,x = 2vw,u¯u,xN(u)vw¯,x\array{ \langle (u v)(w u), x \rangle & = & \langle u v, x(\bar{u}\bar{w})\rangle\\ & \stackrel{Ex}{=} & 2\langle u, x\rangle\langle v, \bar{u}\bar{w} \rangle - \langle u(\bar{u}\bar{w}), x v \rangle \\ & = & 2\langle u, x\rangle\langle v w, \bar{u} \rangle - \langle \bar{u}\bar{w}, \bar{u}(x v) \rangle \\ & = & 2\langle v w, \bar{u} \rangle\langle u, x\rangle - N(u)\langle \bar{w}\bar{v}, x \rangle \\ & = & 2\langle v w, \bar{u} \rangle\langle u, x\rangle - N(u)\langle \widebar{v w}, x \rangle }

which makes it plain that (uv)(wu)(u v)(w u) depends on uu and vwv w only. Hence we get the same result if we replace vv and ww and any two elements whose product is vwv w, say vwv w and ee. In other words,

(uv)(wu)=(u(vw))(eu)=(u(vw))u,(uv)(wu)=(ue)((vw)u)=u((vw)u)(u v)(w u) = (u(v w))(e u) = (u(v w))u, \qquad (u v)(w u) = (u e)((v w)u) = u((v w)u)

which completes the proof.

Corollary

For all uu, vv in a composition algebra, the third alternative law holds: u(vu)=(uv)uu(v u) = (u v)u.

See also Moufang loop.

References

  • Markus Rost, On the dimension of a composition algebra, Documenta Mathematica 1 (1996), 209-214, files Abstract: “The possible dimensions of a composition algebra are 1, 2, 4, or 8. We give a tensor categorical argument.”

An exposition of the string diagram proof of the Hurwitz’ theorem on the classification of compositon algebras is given in

  • Bruce Westbury, Hurwitz’ theorem on composition algebras (arXiv:1011.6197)

  • John Baez’s comment

  • John Conway, Derek A. Smith, On Quaternions and Octonions, A.K. Peters, 2003.

  • T.A. Springer, F.D. Veldkamp, Octonions, Jordan algebras, and exceptional groups, Springer Monographs in Mathematics, Springer-Verlag 2000.

  • wikipedia

Related nnlab entries: alternative algebra, Cayley-Dickson construction

Related eom entries: Lie-admissible algebra

Revised on April 14, 2013 16:46:56 by Todd Trimble (67.81.93.26)