Hurwitz's theorem (composition algebras)

Template:Short description In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a nondegenerate positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions, or the octonions, and that there are no other possibilities. Such algebras, sometimes called Hurwitz algebras, are examples of composition algebras.

The theory of composition algebras has subsequently been generalized to arbitrary quadratic forms and arbitrary fields. Hurwitz's theorem implies that multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originally proved by Hurwitz in 1898. It is a special case of the Hurwitz problem, solved also in Template:Harvtxt. Subsequent proofs of the restrictions on the dimension have been given by Template:Harvtxt using the representation theory of finite groups and by Template:Harvtxt and Template:Harvtxt using Clifford algebras. Hurwitz's theorem has been applied in algebraic topology to problems on vector fields on spheres and the homotopy groups of the classical groups and in quantum mechanics to the classification of simple Jordan algebras.

Euclidean Hurwitz algebras

Definition

A Hurwitz algebra or composition algebra is a finite-dimensional not necessarily associative algebra Template:Mvar with identity endowed with a nondegenerate quadratic form Template:Mvar such that Template:Math. If the underlying coefficient field is the reals and Template:Mvar is positive-definite, so that Template:Math is an inner product, then Template:Mvar is called a Euclidean Hurwitz algebra or (finite-dimensional) normed division algebra.

If Template:Mvar is a Euclidean Hurwitz algebra and Template:Mvar is in Template:Mvar, define the involution and right and left multiplication operators by

$a^{*}=-a+2(a,1)1,L(a)b=ab,R(a)b=ba.$

Evidently the involution has period two and preserves the inner product and norm. These operators have the following properties:

• the involution is an antiautomorphism, i.e. Template:Math
• Template:Math
• Template:Math, Template:Math, so that the involution on the algebra corresponds to taking adjoints
• Template:Math if Template:Math
• Template:Math
• Template:Math, Template:Math, so that Template:Mvar is an alternative algebra.

These properties are proved starting from the polarized version of the identity Template:Math:

$2(a,b)(c,d)=(ac,bd)+(ad,bc).$

Setting Template:Math or Template:Math yields Template:Math and Template:Math.

Hence Template:Math.

Similarly Template:Math.

Hence Template:Math, so that Template:Math.

By the polarized identity Template:Math so Template:Math. Applied to 1 this gives Template:Math. Replacing Template:Mvar by Template:Math gives the other identity.

Substituting the formula for Template:Math in Template:Math gives Template:Math. The formula Template:Math is proved analogously.

Classification

It is routine to check that the real numbers Template:Math, the complex numbers Template:Math and the quaternions Template:Math are examples of associative Euclidean Hurwitz algebras with their standard norms and involutions. There are moreover natural inclusions Template:Math.

Analysing such an inclusion leads to the Cayley–Dickson construction, formalized by A.A. Albert. Let Template:Mvar be a Euclidean Hurwitz algebra and Template:Mvar a proper unital subalgebra, so a Euclidean Hurwitz algebra in its own right. Pick a unit vector Template:Mvar in Template:Mvar orthogonal to Template:Mvar. Since Template:Math, it follows that Template:Math and hence Template:Math. Let Template:Mvar be subalgebra generated by Template:Mvar and Template:Mvar. It is unital and is again a Euclidean Hurwitz algebra. It satisfies the following Cayley–Dickson multiplication laws:

$C=B\oplus Bj,(a+bj{)}^{*}=a^{*}-bj,(a+bj)(c+dj)=(ac-d^{*}b)+(b{c}^{*}+da)j.$

Template:Mvar and Template:Math are orthogonal, since Template:Mvar is orthogonal to Template:Mvar. If Template:Mvar is in Template:Mvar, then Template:Math, since by orthogonal Template:Math. The formula for the involution follows. To show that Template:Math is closed under multiplication Template:Math. Since Template:Math is orthogonal to 1, Template:Math.

• Template:Math since Template:Math so that, for Template:Mvar in Template:Mvar, Template:Math.
• Template:Math taking adjoints above.
• Template:Math since Template:Math = 0, so that, for Template:Mvar in Template:Mvar, Template:Math.

Imposing the multiplicativity of the norm on Template:Mvar for Template:Math and Template:Math gives:

$(\Vert a{\Vert }^2+\Vert b{\Vert }^2)(\Vert c{\Vert }^2+\Vert d{\Vert }^2)=\Vert ac-d^{*}b{\Vert }^2+\Vert b{c}^{*}+da{\Vert }^2,$

which leads to

$(ac,d^{*}b)=(b{c}^{*},da).$

Hence Template:Math, so that Template:Mvar must be associative.

This analysis applies to the inclusion of Template:Math in Template:Math and Template:Math in Template:Math. Taking Template:Math with the product and inner product above gives a noncommutative nonassociative algebra generated by Template:Math. This recovers the usual definition of the octonions or Cayley numbers. If Template:Mvar is a Euclidean algebra, it must contain Template:Math. If it is strictly larger than Template:Math, the argument above shows that it contains Template:Math. If it is larger than Template:Math, it contains Template:Math. If it is larger still, it must contain Template:Math. But there the process must stop, because Template:Math is not associative. In fact Template:Math is not commutative and Template:Math in Template:Math.

Template:Smallcaps The only Euclidean Hurwitz algebras are the real numbers, the complex numbers, the quaternions and the octonions.

Other proofs

The proofs of Template:Harvtxt and Template:Harvtxt use Clifford algebras to show that the dimension Template:Mvar of Template:Mvar must be 1, 2, 4 or 8. In fact the operators Template:Math with Template:Math satisfy Template:Math and so form a real Clifford algebra. If Template:Mvar is a unit vector, then Template:Math is skew-adjoint with square Template:Math. So Template:Mvar must be either even or 1 (in which case Template:Mvar contains no unit vectors orthogonal to 1). The real Clifford algebra and its complexification act on the complexification of Template:Mvar, an Template:Mvar-dimensional complex space. If Template:Mvar is even, Template:Math is odd, so the Clifford algebra has exactly two complex irreducible representations of dimension Template:Math. So this power of 2 must divide Template:Mvar. It is easy to see that this implies Template:Mvar can only be 1, 2, 4 or 8.

The proof of Template:Harvtxt uses the representation theory of finite groups, or the projective representation theory of elementary abelian 2-groups, known to be equivalent to the representation theory of real Clifford algebras. Indeed, taking an orthonormal basis Template:Math of the orthogonal complement of 1 gives rise to operators Template:Math satisfying

$U_i^2=-I,U_i{U}_j=-U_j{U}_i(i\ne j).$

This is a projective representation of a direct product of Template:Math groups of order 2. (Template:Mvar is assumed to be greater than 1.) The operators Template:Math by construction are skew-symmetric and orthogonal. In fact Eckmann constructed operators of this type in a slightly different but equivalent way. It is in fact the method originally followed in Template:Harvtxt. Assume that there is a composition law for two forms

$(x_1^2+\cdots +x_N^2)(y_1^2+\cdots +y_N^2)=z_1^2+\cdots +z_N^2,$

where Template:Math is bilinear in Template:Mvar and Template:Mvar. Thus

$z_i=\sum _{j=1}^N{a}_{ij}(x)y_j$

where the matrix Template:Math is linear in Template:Mvar. The relations above are equivalent to

$T(x)T(x{)}^t=x_1^2+\cdots +x_N^2.$

Writing

$T(x)=T_1{x}_1+\cdots +T_N{x}_N,$

the relations become

$T_i{T}_j^t+T_j{T}_i^t=2{\delta }_{ij}I.$

Now set Template:Math. Thus Template:Math and the Template:Math are skew-adjoint, orthogonal satisfying exactly the same relations as the Template:Math's:

$V_i^2=-I,V_i{V}_j=-V_j{V}_i(i\ne j).$

Since Template:Math is an orthogonal matrix with square Template:Math on a real vector space, Template:Mvar is even.

Let Template:Mvar be the finite group generated by elements Template:Math such that

$v_i^2=\epsilon ,v_i{v}_j=\epsilon v_j{v}_i(i\ne j),$

where Template:Mvar is central of order 2. The commutator subgroup Template:Math is just formed of 1 and Template:Mvar. If Template:Mvar is odd this coincides with the center while if Template:Mvar is even the center has order 4 with extra elements Template:Math and Template:Math. If Template:Mvar in Template:Mvar is not in the center its conjugacy class is exactly Template:Mvar and Template:Math. Thus there are Template:Math conjugacy classes for Template:Mvar odd and Template:Math for Template:Mvar even. Template:Mvar has Template:Math 1-dimensional complex representations. The total number of irreducible complex representations is the number of conjugacy classes. So since Template:Mvar is even, there are two further irreducible complex representations. Since the sum of the squares of the dimensions equals Template:Math and the dimensions divide Template:Math, the two irreducibles must have dimension Template:Math. When Template:Mvar is even, there are two and their dimension must divide the order of the group, so is a power of two, so they must both have dimension Template:Math. The space on which the Template:Math's act can be complexified. It will have complex dimension Template:Mvar. It breaks up into some of complex irreducible representations of Template:Mvar, all having dimension Template:Math. In particular this dimension is Template:Math, so Template:Mvar is less than or equal to 8. If Template:Math, the dimension is 4, which does not divide 6. So N can only be 1, 2, 4 or 8.

Applications to Jordan algebras

Let Template:Mvar be a Euclidean Hurwitz algebra and let Template:Math be the algebra of Template:Mvar-by-Template:Mvar matrices over Template:Mvar. It is a unital nonassociative algebra with an involution given by

$(x_{ij}{)}^{*}=(x_{ji}^{*}).$

The trace Template:Math is defined as the sum of the diagonal elements of Template:Mvar and the real-valued trace by Template:Math. The real-valued trace satisfies:

${\mathrm{Tr}}_{𝐑}XY={\mathrm{Tr}}_{𝐑}YX,{\mathrm{Tr}}_{𝐑}(XY)Z={\mathrm{Tr}}_{𝐑}X(YZ).$

These are immediate consequences of the known identities for Template:Math.

In Template:Mvar define the associator by

$[a,b,c]=a(bc)-(ab)c.$

It is trilinear and vanishes identically if Template:Mvar is associative. Since Template:Mvar is an alternative algebra Template:Math and Template:Math. Polarizing it follows that the associator is antisymmetric in its three entries. Furthermore, if Template:Mvar, Template:Mvar or Template:Mvar lie in Template:Math then Template:Math. These facts imply that Template:Math has certain commutation properties. In fact if Template:Mvar is a matrix in Template:Math with real entries on the diagonal then

$[X,X^2]=aI,$

with Template:Mvar in Template:Mvar. In fact if Template:Math, then

$y_{ij}=\sum _{k,\ell }[x_{ik},x_{k\ell },x_{\ell j}].$

Since the diagonal entries of Template:Mvar are real, the off-diagonal entries of Template:Mvar vanish. Each diagonal entry of Template:Mvar is a sum of two associators involving only off diagonal terms of Template:Mvar. Since the associators are invariant under cyclic permutations, the diagonal entries of Template:Mvar are all equal.

Let Template:Math be the space of self-adjoint elements in Template:Math with product Template:Math and inner product Template:Math.

Template:Smallcaps Template:Math is a Euclidean Jordan algebra if Template:Mvar is associative (the real numbers, complex numbers or quaternions) and Template:Math or if Template:Mvar is nonassociative (the octonions) and Template:Math.

The exceptional Jordan algebra Template:Math is called the Albert algebra after A.A. Albert.

To check that Template:Math satisfies the axioms for a Euclidean Jordan algebra, the real trace defines a symmetric bilinear form with Template:Math. So it is an inner product. It satisfies the associativity property Template:Math because of the properties of the real trace. The main axiom to check is the Jordan condition for the operators Template:Math defined by Template:Math:

$[L(X),L(X^2)]=0.$

This is easy to check when Template:Mvar is associative, since Template:Math is an associative algebra so a Jordan algebra with Template:Math. When Template:Math and Template:Math a special argument is required, one of the shortest being due to Template:Harvtxt.

In fact if Template:Mvar is in Template:Math with Template:Math, then

$D(X)=TX-XT$

defines a skew-adjoint derivation of Template:Math. Indeed,

$\mathrm{Tr}(T(X(X^2))-T(X^2(X)))=\mathrm{Tr}T(aI)=\mathrm{Tr}(T)a=0,$

so that

$(D(X),X^2)=0.$

Polarizing yields:

$(D(X),Y\circ Z)+(D(Y),Z\circ X)+(D(Z),X\circ Y)=0.$

Setting Template:Math shows that Template:Mvar is skew-adjoint. The derivation property Template:Math follows by this and the associativity property of the inner product in the identity above.

With Template:Mvar and Template:Mvar as in the statement of the theorem, let Template:Mvar be the group of automorphisms of Template:Math leaving invariant the inner product. It is a closed subgroup of Template:Math so a compact Lie group. Its Lie algebra consists of skew-adjoint derivations. Template:Harvtxt showed that given Template:Mvar in Template:Mvar there is an automorphism Template:Mvar in Template:Mvar such that Template:Math is a diagonal matrix. (By self-adjointness the diagonal entries will be real.) Freudenthal's diagonalization theorem immediately implies the Jordan condition, since Jordan products by real diagonal matrices commute on Template:Math for any non-associative algebra Template:Mvar.

To prove the diagonalization theorem, take Template:Mvar in Template:Mvar. By compactness Template:Mvar can be chosen in Template:Mvar minimizing the sums of the squares of the norms of the off-diagonal terms of Template:Math. Since Template:Mvar preserves the sums of all the squares, this is equivalent to maximizing the sums of the squares of the norms of the diagonal terms of Template:Math. Replacing Template:Mvar by Template:Math, it can be assumed that the maximum is attained at Template:Mvar. Since the symmetric group Template:Math, acting by permuting the coordinates, lies in Template:Mvar, if Template:Mvar is not diagonal, it can be supposed that Template:Math and its adjoint Template:Math are non-zero. Let Template:Mvar be the skew-adjoint matrix with Template:Math entry Template:Mvar, Template:Math entry Template:Math and 0 elsewhere and let Template:Mvar be the derivation ad Template:Mvar of Template:Mvar. Let Template:Math in Template:Mvar. Then only the first two diagonal entries in Template:Math differ from those of Template:Mvar. The diagonal entries are real. The derivative of Template:Math at Template:Math is the Template:Math coordinate of Template:Math, i.e. Template:Math. This derivative is non-zero if Template:Math. On the other hand, the group Template:Math preserves the real-valued trace. Since it can only change Template:Math and Template:Math, it preserves their sum. However, on the line Template:Math constant, Template:Math has no local maximum (only a global minimum), a contradiction. Hence Template:Mvar must be diagonal.

See also

• Multiplicative quadratic form
• Radon–Hurwitz number
• Frobenius Theorem

References

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Further reading

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• Max Koecher & Reinhold Remmert (1990) "Composition Algebras. Hurwitz's Theorem — Vector-Product Algebras", chapter 10 of Numbers by Heinz-Dieter Ebbinghaus et al., Springer, Template:ISBN
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