Inmathematics,Hurwitz's theorem is atheorem ofAdolf Hurwitz, published posthumously in 1923, solving theHurwitz problem forfinite-dimensionalunitalrealnon-associative algebras endowed with a nondegeneratepositive-definitequadratic form. The theorem states that if the quadratic form defines ahomomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must beisomorphic to thereal numbers, thecomplex numbers, thequaternions, or theoctonions, and that there are no other possibilities. Such algebras, sometimes calledHurwitz algebras, are examples ofcomposition algebras.

The theory of composition algebras has subsequently been generalized to arbitrary quadratic forms and arbitraryfields.^[1] Hurwitz's theorem implies that multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originallyproved by Hurwitz in 1898. It is a special case of theHurwitz problem, solved also inRadon (1922). Subsequent proofs of the restrictions on the dimension have been given byEckmann (1943) using therepresentation theory of finite groups and byLee (1948) andChevalley (1954) usingClifford algebras. Hurwitz's theorem has been applied inalgebraic topology to problems onvector fields on spheres and thehomotopy groups of theclassical groups^[2] and inquantum mechanics to theclassification of simple Jordan algebras.^[3]

AHurwitz algebra orcomposition algebra is a finite-dimensional not necessarily associative algebraA with identity endowed with a nondegenerate quadratic formq such thatq(a b) =q(a) q(b). If the underlying coefficient field is the reals andq is positive-definite, so that(a, b) =⁠1/2⁠[q(a +b) −q(a) −q(b)] is aninner product, thenA is called aEuclidean Hurwitz algebra or (finite-dimensional)normed division algebra.^[4]

IfA is a Euclidean Hurwitz algebra anda is inA, define theinvolution and right and left multiplication operators by

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

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

• the involution is anantiautomorphism, i.e.(ab)* =b*a*
• aa* = ‖a‖² 1 =a*a
• L(a*) =L(a)*,R(a*) =R(a)*, so that the involution on the algebra corresponds to takingadjoints
• Re (ab) = Re (ba) ifRe x = (x +x*)/2 = (x, 1)1
• Re (ab)c = Re a(bc)
• L(a²) =L(a)²,R(a²) =R(a)², so thatA is analternative algebra.

These properties are proved starting from the polarized version of the identity(ab, ab) = (a, a)(b, b):

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

Settingb = 1 ord = 1 yieldsL(a*) =L(a)* andR(c*) =R(c)*.

HenceRe(ab) = (ab, 1)1 = (a, b*)1 = (ba, 1)1 = Re(ba).

SimilarlyRe (ab)c = ((ab)c,1)1 = (ab, c*)1 = (b, a* c*)1 = (bc,a*)1 = (a(bc),1)1 = Rea(bc).

Hence((ab)*, c) = (ab,c*) = (b,a*c*) = (1,b*(a*c*)) = (1, (b*a*)c*) = (b*a*,c), so that(ab)* =b*a*.

By the polarized identity‖a‖² (c, d) = (ac, ad) = (a* (ac), d) soL(a*) L(a) =L(‖a‖²). Applied to 1 this givesa*a = ‖a‖² 1. Replacinga bya* gives the other identity.

Substituting the formula fora* inL(a*) L(a) =L(a*a) givesL(a)² =L(a²). The formulaR(a²) =R(a)² is proved analogously.

It is routine to check that the real numbersR, the complex numbersC and the quaternionsH are examples of associative Euclidean Hurwitz algebras with their standard norms and involutions. There are moreover natural inclusionsR ⊂C ⊂H.

Analysing such an inclusion leads to theCayley–Dickson construction, formalized byA.A. Albert. LetA be a Euclidean Hurwitz algebra andB a proper unital subalgebra, so a Euclidean Hurwitz algebra in its own right. Pick aunit vectorj inA orthogonal toB. Since(j, 1) = 0, it follows thatj* = −j and hencej² = −1. LetC be subalgebra generated byB andj. It is unital and is again a Euclidean Hurwitz algebra. It satisfies the followingCayley–Dickson multiplication laws:

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

B andBj are orthogonal, sincej is orthogonal toB. Ifa is inB, thenj a =a* j, since by orthogonal0 = 2(j, a*) =ja −a*j. The formula for the involution follows. To show thatB ⊕B j is closed under multiplicationBj =jB. SinceBj is orthogonal to 1,(bj)* = −bj.

• b(cj) = (cb)j since(b, j) = 0 so that, forx inA,(b(cj), x) = (b(jx), j(cj)) = −(b(jx), c*) = −(cb, (jx)*) = −((cb)j, x*) = ((cb)j, x).
• (jc)b =j(bc) taking adjoints above.
• (bj)(cj) = −c*b since(b, cj) = 0, so that, forx inA,((bj)(cj), x) = −((cj)x*, bj) = (bx*, (cj)j) = −(c*b, x).

Imposing the multiplicativity of the norm onC fora +bj andc +dj gives:

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

which leads to

${\displaystyle {\displaystyle (ac,d^{\ast }b)=(b{c}^{\ast },da).}}$

Henced(ac) = (da)c, so thatBmust be associative.

This analysis applies to the inclusion ofR inC andC inH. TakingO =H ⊕H with the product and inner product above gives a noncommutative nonassociative algebra generated byJ = (0, 1). This recovers the usual definition of theoctonions orCayley numbers. IfA is a Euclidean algebra, it must containR. If it is strictly larger thanR, the argument above shows that it containsC. If it is larger thanC, it containsH. If it is larger still, it must containO. But there the process must stop, becauseO is not associative. In factH is not commutative anda(bj) = (ba)j ≠ (ab)j inO.^[5]

Theorem. The only Euclidean Hurwitz algebras are the real numbers, the complex numbers, the quaternions and the octonions.

The proofs ofLee (1948) andChevalley (1954) useClifford algebras to show that the dimensionN ofA must be 1, 2, 4 or 8. In fact the operatorsL(a) with(a, 1) = 0 satisfyL(a)² = −‖a‖² and so form a real Clifford algebra. Ifa is a unit vector, thenL(a) is skew-adjoint with square−I. SoN must be eithereven or 1 (in which caseA contains no unit vectors orthogonal to 1). The real Clifford algebra and itscomplexification act on the complexification ofA, anN-dimensional complex space. IfN is even,N − 1 isodd, so the Clifford algebra has exactly two complexirreducible representations of dimension2^{N/2 − 1}. So thispower of 2 must divideN. It is easy to see that this impliesN can only be 1, 2, 4 or 8.

The proof ofEckmann (1943) uses therepresentation theory offinite groups, or the projective representation theory ofelementary abelian 2-groups, known to be equivalent to the representation theory of real Clifford algebras. Indeed, taking anorthonormal basise_i of theorthogonal complement of 1 gives rise to operatorsU_i =L(e_i)satisfying

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

This is aprojective representation of a direct product ofN − 1groups oforder 2. (N is assumed to be greater than 1.) The operatorsU_i 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 inHurwitz (1923).^[6] Assume that there is a composition law for two forms

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

wherez_i is bilinear inx andy. Thus

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

where thematrixT(x) = (a_ij) is linear inx. The relations above are equivalent to

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

Writing

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

the relations become

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

Now setV_i = (T_N)^tT_i. ThusV_N =I and theV₁, ... , V_{N − 1} are skew-adjoint, orthogonal satisfying exactly the same relations as theU_i's:

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

SinceV_i is anorthogonal matrix with square−I on a realvector space,N is even.

LetG be the finite group generated by elementsv_i such that

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

whereε iscentral of order 2. Thecommutator subgroup[G, G] is just formed of 1 andε. IfN is odd this coincides with thecenter while ifN is even the center has order 4 with extra elementsγ =v₁...v_{N − 1} andεγ. Ifg inG is not in the center itsconjugacy class is exactlyg andεg. Thus there are2^{N − 1} + 1 conjugacy classes forN odd and2^{N − 1} + 2 forN even.G has|G / [G, G] | = 2^{N − 1} 1-dimensional complex representations. The total number of irreducible complex representations is the number of conjugacy classes. So sinceN is even, there are two further irreducible complex representations. Since the sum of the squares of the dimensions equals|G| and the dimensions divide|G|, the two irreducibles must have dimension2^{(N − 2)/2}. WhenN 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 dimension2^{(N − 2)/2}. The space on which theV_i's act can be complexified. It will have complex dimensionN. It breaks up into some of complex irreducible representations ofG, all having dimension2^{(N − 2)/2}. In particular this dimension is≤N, soN is less than or equal to 8. IfN = 6, the dimension is 4, which does not divide 6. SoN can only be 1, 2, 4 or 8.

LetA be a Euclidean Hurwitz algebra and letM_n(A) be the algebra ofn-by-n matrices overA. It is a unital nonassociative algebra with an involution given by

${\displaystyle {\displaystyle (x_{ij}{)}^{\ast }=(x_{ji}^{\ast }).}}$

The traceTr(X) is defined as the sum of the diagonal elements ofX and the real-valued trace byTr_R(X ) = Re Tr(X ). The real-valued trace satisfies:

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

These are immediate consequences of the known identities forn = 1.

InA define theassociator by

${\displaystyle {\displaystyle [a,b,c]=a(bc)-(ab)c.}}$

It is trilinear and vanishes identically ifA is associative. SinceA is analternative algebra[a, a, b] = 0 and[b, a, a] = 0. Polarizing it follows that the associator is antisymmetric in its three entries. Furthermore, ifa,b orc lie inR then[a, b, c] = 0. These facts imply thatM₃(A) has certain commutation properties. In fact ifX is a matrix inM₃(A) with real entries on the diagonal then

${\displaystyle {\displaystyle [X,X^2]=aI,}}$

witha inA. In fact ifY = [X,  X²], then

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

Since the diagonal entries ofX are real, the off-diagonal entries ofY vanish. Each diagonalentry ofY is a sum of two associators involving only off diagonal terms ofX. Since the associators are invariant undercyclic permutations, the diagonal entries ofY are all equal.

LetH_n(A) be the space of self-adjoint elements inM_n(A) with productX ∘Y =⁠1/2⁠(X Y +Y X) and inner product(X, Y ) = Tr_R(X Y ).

Theorem.H_n(A) is aEuclidean Jordan algebra ifA is associative (the real numbers, complex numbers or quaternions) andn ≥ 3 or ifA is nonassociative (the octonions) andn = 3.

Theexceptional Jordan algebraH₃(O) is called theAlbert algebra afterA.A. Albert.

To check thatH_n(A) satisfies the axioms for a Euclidean Jordan algebra, the real trace defines asymmetric bilinear form with(X, X) = Σ ‖x_ij‖². So it is an inner product. It satisfies the associativity property(Z∘X, Y ) = (X, Z∘Y ) because of the properties of the real trace. The main axiom to check is the Jordan condition for the operatorsL(X) defined byL(X)Y =X ∘Y:

${\displaystyle {\displaystyle [L(X),L(X^2)]=0.}}$

This is easy to check whenA is associative, sinceM_n(A) is an associative algebra so a Jordan algebra withX ∘Y =⁠1/2⁠(X Y +Y X). WhenA =O andn = 3 a special argument is required, one of the shortest being due toFreudenthal (1951).^[7]

In fact ifT is inH₃(O) withTr T = 0, then

${\displaystyle {\displaystyle D(X)=TX-XT}}$

defines a skew-adjoint derivation ofH₃(O). Indeed,

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

so that

${\displaystyle (D(X),X^2)=0.}$

Polarizing yields:

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

SettingZ = 1 shows thatD is skew-adjoint. The derivation propertyD(X ∘Y) =D(X)∘Y +X∘D(Y) follows by this and the associativity property of the inner product in the identity above.

WithA andn as in the statement of the theorem, letK be the group ofautomorphisms ofE =H_n(A) leaving invariant the inner product. It is a closed subgroup ofO(E) so acompactLie group. ItsLie algebra consists of skew-adjoint derivations.Freudenthal (1951) showed that givenX inE there is an automorphismk inK such thatk(X) is adiagonal 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 onM_n(A) for any non-associative algebraA.

To prove the diagonalization theorem, takeX inE. By compactnessk can be chosen inK minimizing the sums of the squares of the norms of the off-diagonal terms ofk(X ). SinceK preserves the sums of all the squares, this is equivalent to maximizing the sums of the squares of the norms of the diagonal terms ofk(X ). ReplacingX byk X, it can be assumed that the maximum is attained atX. Since thesymmetric groupS_n, acting by permuting the coordinates, lies inK, ifX is not diagonal, it can be supposed thatx₁₂ and its adjointx₂₁ are non-zero. LetT be the skew-adjoint matrix with(2, 1) entrya,(1, 2) entry−a* and 0 elsewhere and letD be the derivation adT ofE. Letk_t = exp tD inK. Then only the first two diagonal entries inX(t) =k_tX differ from those ofX. The diagonal entries are real. The derivative ofx₁₁(t) att = 0 is the(1, 1) coordinate of[T, X], i.e.a* x₂₁ +x₁₂ a = 2(x₂₁, a). This derivative is non-zero ifa =x₂₁. On the other hand, the groupk_t preserves the real-valued trace. Since it can only changex₁₁ andx₂₂, it preserves their sum. However, on the linex +y = constant,x² +y² has no local maximum (only a global minimum), a contradiction. HenceX must be diagonal.

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

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• Composition algebras
• Non-associative algebras
• Quadratic forms
• Representation theory
• Theorems about algebras
• 1923 introductions
• Hypercomplex numbers
• Octonions

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