In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz, 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 Radon (1922). Subsequent proofs of the restrictions on the dimension have been given by Eckmann (1943) using the representation theory of finite groups and by Lee (1948) and Chevalley (1954) 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 A with identity endowed with a nondegenerate quadratic form q such that q(a b) = q(a) q(b). If the underlying coefficient field is the reals and q is positive-definite, so that (a, b) = 1/2[q(a + b) − q(a) − q(b)] is an inner product, then A is called a Euclidean Hurwitz algebra or (finite-dimensional) normed division algebra. If A is a Euclidean Hurwitz algebra and a is in A, define the involution and right and left multiplication operators by
a ∗ = − a + 2 ( a , 1 ) 1 , L ( a ) b = a b , R ( a ) b = b a . {\displaystyle a^{*}=-a+2(a,1)1,\quad L(a)b=ab,\quad 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. (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 taking adjoints Re (ab) = Re (ba) if Re x = (x + x*)/2 = (x, 1)1 Re (ab)c = Re a(bc) L(a2) = L(a)2, R(a2) = R(a)2, so that A is an alternative algebra. These properties are proved starting from the polarized version of the identity (ab, ab) = (a, a)(b, b):
2 ( a , b ) ( c , d ) = ( a c , b d ) + ( a d , b c ) . {\displaystyle \displaystyle {2(a,b)(c,d)=(ac,bd)+(ad,bc).}}
Setting b = 1 or d = 1 yields L(a*) = L(a)* and R(c*) = R(c)*. Hence Re(ab) = (ab, 1)1 = (a, b*)1 = (ba, 1)1 = Re(ba). Similarly Re (ab)c = ((ab)c,1)1 = (ab, c*)1 = (b, a* c*)1 = (bc,a*)1 = (a(bc),1)1 = Re a(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) so L(a*) L(a) = L(‖a‖2). Applied to 1 this gives a*a = ‖a‖2 1. Replacing a by a* gives the other identity. Substituting the formula for a* in L(a*) L(a) = L(a*a) gives L(a)2 = L(a2). The formula R(a2) = R(a)2 is proved analogously.
Classification It is routine to check that the real numbers R, the complex numbers C and the quaternions H are examples of associative Euclidean Hurwitz algebras with their standard norms and involutions. There are moreover natural inclusions R ⊂ C ⊂ H. Analysing such an inclusion leads to the Cayley–Dickson construction, formalized by A.A. Albert. Let A be a Euclidean Hurwitz algebra and B a proper unital subalgebra, so a Euclidean Hurwitz algebra in its own right. Pick a unit vector j in A orthogonal to B. Since (j, 1) = 0, it follows that j* = −j and hence j2 = −1. Let C be subalgebra generated by B and j. It is unital and is again a Euclidean Hurwitz algebra. It satisfies the following Cayley–Dickson multiplication laws:
C = B ⊕ B j , ( a + b j ) ∗ = a ∗ − b j , ( a + b j ) ( c + d j ) = ( a c − d ∗ b ) + ( b c ∗ + d a ) j . {\displaystyle \displaystyle {C=B\oplus Bj,\,\,\,(a+bj)^{*}=a^{*}-bj,\,\,\,(a+bj)(c+dj)=(ac-d^{*}b)+(bc^{*}+da)j.}}
B and Bj are orthogonal, since j is orthogonal to B. If a is in B, then j a = a* j, since by orthogonal 0 = 2(j, a*) = ja − a*j. The formula for the involution follows. To show that B ⊕ B j is closed under multiplication Bj = jB. Since Bj is orthogonal to 1, (bj)* = −bj.
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