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.[1] 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 . Subsequent proofs of the restrictions on the dimension have been given by using the representation theory of finite groups and by and 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[2] and in quantum mechanics to the classification of simple Jordan algebras.
A Hurwitz algebra or composition algebra is a finite-dimensional not necessarily associative algebra with identity endowed with a nondegenerate quadratic form such that . If the underlying coefficient field is the reals and is positive-definite, so that is an inner product, then is called a Euclidean Hurwitz algebra or (finite-dimensional) normed division algebra.
If is a Euclidean Hurwitz algebra and is in, 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:
These properties are proved starting from the polarized version of the identity :
\displaystyle{2(a,b)(c,d)=(ac,bd)+(ad,bc).}
Setting or yields and .
Hence .
Similarly .
Hence, so that .
By the polarized identity so . Applied to 1 this gives . Replacing by gives the other identity.
Substituting the formula for in gives . The formula is proved analogously.
It is routine to check that the real numbers, the complex numbers and the quaternions are examples of associative Euclidean Hurwitz algebras with their standard norms and involutions. There are moreover natural inclusions .
Analysing such an inclusion leads to the Cayley–Dickson construction, formalized by A.A. Albert. Let be a Euclidean Hurwitz algebra and a proper unital subalgebra, so a Euclidean Hurwitz algebra in its own right. Pick a unit vector in orthogonal to . Since, it follows that and hence . Let be subalgebra generated by and . It is unital and is again a Euclidean Hurwitz algebra. It satisfies the following Cayley–Dickson multiplication laws:
\displaystyle{C=B ⊕ Bj,(a+bj)*=a*-bj,(a+bj)(c+dj)=(ac-d*b)+(bc*+da)j.}
and are orthogonal, since is orthogonal to . If is in, then, since by orthogonal . The formula for the involution follows. To show that is closed under multiplication . Since is orthogonal to 1, .
Imposing the multiplicativity of the norm on for and gives:
\displaystyle{(\|a\|2+\|b\|2)(\|c\|2+\|d\|2)=\|ac-d*b\|2+\|bc*+da\|2,}
which leads to
\displaystyle{(ac,d*b)=(bc*,da).}
Hence, so that must be associative.
This analysis applies to the inclusion of in and in . Taking with the product and inner product above gives a noncommutative nonassociative algebra generated by . This recovers the usual definition of the octonions or Cayley numbers. If is a Euclidean algebra, it must contain . If it is strictly larger than, the argument above shows that it contains . If it is larger than, it contains . If it is larger still, it must contain . But there the process must stop, because is not associative. In fact is not commutative and in .
The only Euclidean Hurwitz algebras are the real numbers, the complex numbers, the quaternions and the octonions.
The proofs of and use Clifford algebras to show that the dimension of must be 1, 2, 4 or 8. In fact the operators with satisfy and so form a real Clifford algebra. If is a unit vector, then is skew-adjoint with square . So must be either even or 1 (in which case contains no unit vectors orthogonal to 1). The real Clifford algebra and its complexification act on the complexification of, an -dimensional complex space. If is even, is odd, so the Clifford algebra has exactly two complex irreducible representations of dimension . So this power of 2 must divide . It is easy to see that this implies can only be 1, 2, 4 or 8.
The proof of 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 of the orthogonal complement of 1 gives rise to operators satisfying
| 2 | |
| U | |
| i |
=-I, UiUj=-UjUi(i\nej).
This is a projective representation of a direct product of groups of order 2. (is assumed to be greater than 1.) The operators 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 .[3] Assume that there is a composition law for two forms
| 2 | |
| \displaystyle{(x | |
| 1 |
+ …
| 2 | |
| +x | |
| 1 |
+ … +
| 2) | |
| y | |
| N |
| 2 | |
| =z | |
| 1 |
+ … +
| 2,} | |
| z | |
| N |
where is bilinear in and . Thus
\displaystyle{zi=\sum
| N | |
| j=1 |
aij(x)yj}
where the matrix is linear in . The relations above are equivalent to
| 2 | |
| \displaystyle{T(x)T(x) | |
| 1 |
+ … +
| 2.} | |
| x | |
| N |
Writing
\displaystyle{T(x)=T1x1+ … +TNxN,}
the relations become
| t | |
| \displaystyle{T | |
| j+T |
jT
| t | |
| i |
=2\deltaijI.}
Now set . Thus and the are skew-adjoint, orthogonal satisfying exactly the same relations as the 's:
| 2 | |
| \displaystyle{V | |
| i |
=-I, ViVj=-VjVi(i\nej).}
Since is an orthogonal matrix with square on a real vector space, is even.
Let be the finite group generated by elements such that
| 2 | |
| \displaystyle{v | |
| i |
=\varepsilon, vivj=\varepsilonvjvi(i\nej),}
where is central of order 2. The commutator subgroup is just formed of 1 and . If is odd this coincides with the center while if is even the center has order 4 with extra elements and . If in is not in the center its conjugacy class is exactly and . Thus there are conjugacy classes for odd and for even. has 1-dimensional complex representations. The total number of irreducible complex representations is the number of conjugacy classes. So since is even, there are two further irreducible complex representations. Since the sum of the squares of the dimensions equals and the dimensions divide, the two irreducibles must have dimension . When 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 . The space on which the 's act can be complexified. It will have complex dimension . It breaks up into some of complex irreducible representations of, all having dimension . In particular this dimension is, so is less than or equal to 8. If, the dimension is 4, which does not divide 6. So N can only be 1, 2, 4 or 8.
Let be a Euclidean Hurwitz algebra and let be the algebra of -by- matrices over . It is a unital nonassociative algebra with an involution given by
\displaystyle{(xij
| *).} | |
| ) | |
| ji |
The trace is defined as the sum of the diagonal elements of and the real-valued trace by. The real-valued trace satisfies:
\operatorname{Tr}RXY=\operatorname{Tr}RYX, \operatorname{Tr}R(XY)Z=\operatorname{Tr}RX(YZ).
These are immediate consequences of the known identities for .
In define the associator by
\displaystyle{[a,b,c]=a(bc)-(ab)c.}
It is trilinear and vanishes identically if is associative. Since is an alternative algebra and . Polarizing it follows that the associator is antisymmetric in its three entries. Furthermore, if, or lie in then . These facts imply that has certain commutation properties. In fact if is a matrix in with real entries on the diagonal then
\displaystyle{[X,X2]=aI,}
with in . In fact if, then
\displaystyle{yij=\sumk,\ell[xik,xk\ell,x\ell].}
Since the diagonal entries of are real, the off-diagonal entries of vanish. Each diagonalentry of is a sum of two associators involving only off diagonal terms of . Since the associators are invariant under cyclic permutations, the diagonal entries of are all equal.
Let be the space of self-adjoint elements in with product and inner product .
is a Euclidean Jordan algebra if is associative (the real numbers, complex numbers or quaternions) and or if is nonassociative (the octonions) and .
The exceptional Jordan algebra is called the Albert algebra after A.A. Albert.
To check that satisfies the axioms for a Euclidean Jordan algebra, the real trace defines a symmetric bilinear form with . So it is an inner product. It satisfies the associativity property because of the properties of the real trace. The main axiom to check is the Jordan condition for the operators defined by :
\displaystyle{[L(X),L(X2)]=0.}
This is easy to check when is associative, since is an associative algebra so a Jordan algebra with . When and a special argument is required, one of the shortest being due to .[4]
In fact if is in with, then
\displaystyle{D(X)=TX-XT}
defines a skew-adjoint derivation of . Indeed,
\operatorname{Tr}(T(X(X2))-T(X2(X)))=\operatorname{Tr}T(aI)=\operatorname{Tr}(T)a=0,
so that
(D(X),X2)=0.
Polarizing yields:
(D(X),Y\circZ)+(D(Y),Z\circX)+(D(Z),X\circY)=0.
Setting shows that is skew-adjoint. The derivation property follows by this and the associativity property of the inner product in the identity above.
With and as in the statement of the theorem, let be the group of automorphisms of leaving invariant the inner product. It is a closed subgroup of so a compact Lie group. Its Lie algebra consists of skew-adjoint derivations. showed that given in there is an automorphism in such that 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 for any non-associative algebra .
To prove the diagonalization theorem, take in . By compactness can be chosen in minimizing the sums of the squares of the norms of the off-diagonal terms of . Since 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 . Replacing by, it can be assumed that the maximum is attained at . Since the symmetric group, acting by permuting the coordinates, lies in, if is not diagonal, it can be supposed that and its adjoint are non-zero. Let be the skew-adjoint matrix with entry, entry and 0 elsewhere and let be the derivation ad of . Let in . Then only the first two diagonal entries in differ from those of . The diagonal entries are real. The derivative of at is the coordinate of, i.e. . This derivative is non-zero if . On the other hand, the group preserves the real-valued trace. Since it can only change and, it preserves their sum. However, on the line constant, has no local maximum (only a global minimum), a contradiction. Hence must be diagonal.