Hurwitz's theorem (composition algebras)
Introduction 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 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. 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 . 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 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  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: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. * * , , so that the involution on the algebra corresponds to taking adjoints*  if * * , , so that  is an alternative algebra.These properties are proved starting from the polarized version of the identity ::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.===Classification===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''':: 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, . *  since  so that, for  in , . *  taking adjoints above. *  since  = 0, so that, for  in , .Imposing the multiplicativity of the norm on  for  and  gives::which leads to: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.
Other proofs 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: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 . Assume that there is a composition law for two forms:where  is bilinear in  and . Thus:where the matrix  is linear in . The relations above are equivalent to:Writing:the relations become:Now set . Thus  and the  are skew-adjoint, orthogonal satisfying exactly the same relations as the 's::Since  is an orthogonal matrix with square  on a real vector space,  is even.Let  be the finite group generated by elements  such that: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.
Applications to Jordan algebras 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:The trace  is defined as the sum of the diagonal elements of  and the real-valued trace by. The real-valued trace satisfies::These are immediate consequences of the known identities for .In  define the ''associator'' by: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:with  in . In fact if , then: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 ::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 .In fact if  is in  with , then:defines a skew-adjoint derivation of . Indeed,:so that:Polarizing yields::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.
See also * Multiplicative quadratic form* Radon–Hurwitz number* Frobenius Theorem
Notes
References **** * *** (reprint of 1951 article)****** ** *** * *
Further reading ** ** Max Koecher & Reinhold Remmert (1990) "Composition Algebras. Hurwitz's Theorem — Vector-Product Algebras", chapter 10 of ''Numbers'' by Heinz-Dieter Ebbinghaus et al., Springer, *