composition algebra

1 Definition

The classical definition of a composition algebra is anon-associative algebra $C$ over a field $k$ where

1. $C$ admits a non-degenerate quadratic form $q:C\to k$, suchthat

2. $q$ is multiplicative: $q(xy)=q(x)q(y)$.

We also say $q$ permits composition or that it obeys the composition law.

This definition is geometric in that quadratic forms give rise to geometric atributesfor a vector space such as length, distance and orthogonality. Indeed, originally createdover the real numbers such properties seem appropriate for an algebra; however, conceptsof length and distance are less appropriate over arbitrary fields and encourage a secondequivalent definition based solely on the algebraic aspect of such algebras.

Alternatively, a composition algebra can be defined as a unital alternative algebra $C$over a field $k$ with an involution $x\mapsto \overline{x}$, that is an anti-isomorphismof order at most 2, such that:

1. $C$ has no non-zero absolute zero divisors (that is, $(xa)x=0$ for all $a\in C$ implies $x=0$);

2. the norm $N(x):=x\overline{x}$ is a scalar multiple of $1$, that is, $N:C\to k1$.

The first definition makes the composition property part of the definition but obscuresthe alternative multiplication as well as the existence of an involutary anti-isomorphismfor the algebra. The second definition makes both of these properties evident but obscuresthe composition property of the norm, and also hides the property that $N$ is a quadratic form.However both definitions have merit, the first captures the classical view of an algebrarespecting a certain geometric condition while the second, introduced by Jacobson, promotesa purely algebraic treatment. In our examples and constructions to follows we attempt to exhibitboth aspects by supplying the norm, the involution, and the product.

Both definitions can be generalized to algebras over commutative unital rings $k$.

Recall that a quadratic form gives rise to a symmetric bilinear from $b:C\times C\to k$ by$b(x,y)=q(x+y)-q(x)-q(y)$, for all $x,y\in C$. Some of the immediate properties include:

1. $b(x,x)=2q(x)$,

2. $b(xz,yz)=b(x,y)q(z)$,

3. $b(xy,zw)+b(xw,zy)=b(x,y)b(z,w)$.

These strongly limit the structure of composition algebrasand leads to the celebrated theorem of Hurwitz (see Theorem 4)which suitably classifies the composition algebras over $ℝ$. The work ofmany others including Albert, Dickson, Jacobson, and Kaplansky extended the essentialconclussion of Hurwitz to all fields and the resulting generalization is stillrefered to as Hurwitz’s theorem.

There are other algebras $A$ with norms $q:A\to k$ which permit compositionin the sense that $q(xy)=q(x)q(y)$. For example, alternative algebras with involutions.However, the distinguishing property of composition algebras is that $q$ is a quadratic form.Classifications for such norms have been caried out by Schafer and McCrimmon.

2 Norms

Originally, composition algebras were created over the real numbers $k=ℝ$.Here the usual positive definite norm on the real vector space was used instead ofthe quadratic form (the square of the norm is the quadratic form).

The first non-trivial example is the set of complex numbers $ℂ$ with where$z=a+bi\in ℂ$ is assigned:

$\overline{z}$$=$$a-bi;$

$|z|$$=$$|a+bi|=\sqrt{a^2+b^2}=\sqrt{z\overline{z}}.$

More interesting is the non-commutative algebra of Hamiltonians$ℍ$, created by Hamilton, where each $x\in \mathscr{H}$ has the form$x=a+bi+cj+dk$ and

$\overline{x}$$=$$a-bi-cj-dk;$

$|x|$$=$$|a+bi+cj+dk|=\sqrt{a^2+b^2+c^2+d^2}=\sqrt{x\overline{x}}.$

The last addition to the list was the non-associative algebra of octonionsinitially created by Cayley and the norm is simply

$\overline{x}$$=$$a-bi-cj-dk-fil-gjl-hkl;$

$|x|$$=$$|a+bi+cj+dk+el+fil+gjl+hkl|$

$=$$\sqrt{a^2+b^2+c^2+d^2+e^2+f^2+g^2+h^2}=\sqrt{x\overline{x}}.$

Because general fields do not sufficient squareroots, the use of normsin the classical Euclidean sense is replaced by the use of quadratic forms.Furthermore, the lack of ordering a field, such as a finite field, introduces theneed to use non-degenerate rather than positive definite conditions. Under thesegeneralizations composition algebras can be redefined form the classical contextof composition algebras over $ℝ$ to general composition algebras overarbitrary fields, as done by our original definitions above. In this context,there are three further composition algebras over $ℝ$.

Example 1.

Let $C\mathrm{=}\mathrm{R}\mathrm{\oplus }\mathrm{R}$ with $q\mathit{}\mathrm{(}x\mathrm{,}y\mathrm{)}\mathrm{=}x\mathit{}y$ for all$\mathrm{(}x\mathrm{,}y\mathrm{)}\mathrm{\in }C$. Then $C$ is a composition algebra.

Proof.

Evidently $q(ax,ay)=a^{2}q(x,y)$ and the polarization of $q$ is the symmetric bilinearform $b((x,y),(z,w))=xz-yw$ for all $(x,y),(z,w)\in C$ (so the signature is $(1,-1)$).Thus $q$ is a quadratic form.

To check that $q$ has the compositional property let $(x,y),(z,w)\in C$. Then

$$q((x,y)(z,w))=q(xz,yw)=(xz)(yw)=(xy)(zw)=q(x,y)q(z,w).$$

Note also that by defining $\overline{(x,y)}=(y,x)$ then $(x,y)\overline{(x,y)}=(xy,yx)=q(x,y)(1,1)$ and $b((x,y),(z,w))(1,1)=(x,y)(z,w)+\overline{(x,y)(z,w)}$.∎

Example 2.

Let $C\mathrm{=}{M}_{\mathrm{2}}\mathit{}\mathrm{(}\mathrm{R}\mathrm{)}$ with $q\mathit{}\mathrm{(}X\mathrm{)}\mathrm{=}\det \mathit{}X$ for all $X\mathrm{\in }C$.Then $C$ is a composition algebra.

Proof.

Let $X\in C$ and $a\in ℝ$. Then $q(aX)=det(aX)=det(a{I}_2)detX=a^{2}detX=a^{2}q(X)$.It is also evident that if thensetting makes $(detX)I_2=X\overline{X}$and also $\mathrm{T}(X)I_2=X+\overline{X}$, where $\mathrm{T}(X)$ is the trace of $X$. Hence

$$\begin{array}{c}b(X,Y)=(q(X+Y)-q(X)-q(Y))I_2=(det(X+Y))I_2-(detX)I_2-(detY)I_2\hfill \\ \hfill =(X+Y)\overline{(X+Y)}-X\overline{X}-Y\overline{Y}\\ \hfill =Y\overline{X}+X\overline{Y}=\mathrm{T}(X\overline{Y})I_2.\end{array}$$

Therefore, $b(X,Y)=\mathrm{T}(X\overline{Y})$. Since $\mathrm{T}(X\overline{Y})=\mathrm{T}(\overline{Y}X)=\mathrm{T}(X\overline{Y})$, it followsthat $b$ is a symmetric bilinear form and so $q$ is quadratic form.

Finally, for composition note

$$q(XY)=det(XY)=detXdetY=q(X)q(Y).$$

Therefore $C$ is a composition algebra.∎

This gives two new composition algebras over $ℝ$ and indeed there is a third, constructedbelow as the algebra $\left(\frac{1,1,1}{ℝ}\right)$, which is 8-dimensional and non-associative butunlike the octonions, it has non-trivial zero-divisors.

Definition 3.

A composition algebra is split if the quadratic form is isotropic.

The example of $ℝ\oplus ℝ$ and $M_2(ℝ)$ just given are both examples ofsplit composition algebras.

3 Involution

Define

$$\overline{x}:=b(x,1)1-x,\forall x\in C.$$

Immediately it follows that: for all $x,y\in C$,

1. $\overline{\overline{x}}=x$,

2. $\overline{x+y}=\overline{x}+\overline{y}$,

3. $\overline{1}=1$.

Define the trace of $x$ as $T(x)=x+\overline{x}$ andthe norm of $x$ as $N(x)=\overline{x}x$. Then it follows that:

$$x^2-T(x)x+N(x)=0,\forall x\in C.$$

So $C$ is a quadratic algebra since every element in $x$ has at mosta quadratic minimal polynomial. In fact $N(x)$ is a quadratic formallowing composition.

4 Constructing composition algebras

All of the following are composition algebras.[5, III.4]

$dim1$:

$k$, with trivial involution $x=\overline{x}$ for all $x$ in $k$.

$dim2$:

For any $\alpha \in k$, a quadratic extension of $k$, that is

$$\left(\frac{\alpha }{k}\right)=⟨1,i|i^2=\alpha ⟩.$$

Here $\{1,i\}$ is a basis and has an involution defined by $\overline{1}=1$ and $\overline{i}=-i$.

$dim4$:

For any $\alpha ,\beta \in k$, a quaternion algebra over $k$defined as

$$\left(\frac{\alpha ,\beta }{k}\right)=⟨1,i,j|i^2=\alpha ,j^2=\beta ,ij=-ji⟩$$

Then $\{1,i,j,ij\}$ forms a basis. An involution is defined by $\overline{1}=1$,$\overline{i}=-i$, $\overline{j}=-j$ and extended linearly.

$dim8$:

For any $\alpha ,\beta ,\gamma \in k$, an octonion algebra over $k$:

$$\begin{array}{c}\left(\frac{\alpha ,\beta ,\gamma }{k}\right)=⟨1,i,j,l|i^2=\alpha ,j^2=\beta ,l^2=\gamma ,ij=-ji,il=-li,jl=-lj,\hfill \\ \hfill i(lj)=-l(ij),(li)j=l(ji),(li)(lj)=-\gamma ji⟩.\end{array}$$

The set $\{1,i,j,ij,l,il,jl,ijl\}$ is a basis.An involution is defined by $\overline{1}=1$, $\overline{i}=-i$, $\overline{j}=-j$, $\overline{l}=-l$and extended linearly.

Each of these algebras can be realized by the Cayley-Dickson method which takes $C$an associative $k$-algebra with involution and produces for each $\alpha \in C-\{0\}$ a newalgebra $\left(\frac{\alpha }{C}\right)$ on the vector space $C\oplus C$ with product

$$(a,b)(c,d)=(ac+\alpha d\overline{b},\overline{a}d+cb).$$

Set the involution on $\left(\frac{\alpha }{C}\right)$ to be $\overline{(a,b)}=(\overline{a},-b)$.

The algebras are equipped with a trace $Tr(x)=x+\overline{x}$,and norm $N(x)=x\overline{x}$. This norm serves as the quadratic map to establishthese algebras a composition algebras. The images of the trace and norm lie in $k$.

The new algebra is associative only if $C$ is commutative, otherwise it is alternative.This means that $k,\left({\displaystyle \frac{\alpha }{k}}\right),\left({\displaystyle \frac{\alpha ,\beta }{k}}\right)$ arethe associative composition algebras.

An algebra is a division algebra if the only zero-divisor is $0$[5, II.2]. A central simple composition algebra with a non-trivialzero-divisor is called a split composition algebra. Finite dimensional splitcentral simple composition algebras are unique up to isomorphism to one of

$$k,\left(\frac{1}{k}\right)\cong k\oplus k,\left(\frac{1,1}{k}\right)\cong M_2(k),\left(\frac{1,1,1}{k}\right).$$

5 Classification theorem

Theorem 4.

[2, Theorem 6.2.3]A composition algebra $C$ over a field $k$ with quadratic form $q\mathit{}\mathrm{(}x\mathrm{)}\mathrm{=}x\mathit{}\overline{x}$ is isomorphicto one of the following:

1. A purely inseparable extension field $E/F$ of characteristic $2$ and exponent $1$ (trivialinvolution) so $q(x)=x^2$.

2. $k$ with trivial involution, so $q(x)=x^2$,

3. Quadratic composition algebra: $\left(\frac{\alpha }{k}\right)$ for $\alpha \in k$,

4. Quaternion algebra: $\left(\frac{\alpha ,\beta }{k}\right)$ for $\alpha ,\beta \in k$,

5. Octonion algebra: $\left(\frac{\alpha ,\beta ,\gamma }{k}\right)$ for $\alpha ,\beta ,\gamma \in k$.

In particular, all composition algebras over $k$, save perhaps those of type $\mathrm{(}i\mathrm{)}$, are finitedimensional and of dimension $\mathrm{1}$, $\mathrm{2}$, $\mathrm{4}$ or $\mathrm{8}$.