In mathematics, an octonionalgebra or Cayley algebra over a fieldF is an algebraic structure which is an 8-dimensional composition algebra over F. In other words, it is a unitalnon-associative algebraA over F with a non-degeneratequadratic formN such that for all x and y in A. The most well-known example of an octonion algebra is the classical octonions, which are an octonion algebra over R, the field of real numbers. The split-octonions also form an octonion algebra over R. Up to R-algebra isomorphism, these are the only octonion algebras over the reals. The algebra of bioctonions is the octonion algebra over the complex numbersC. The octonion algebra for N is a division algebraif and only if the form N is anisotropic. A split octonion algebra is one for which the quadratic form N is isotropic. Up to F-algebra isomorphism, there is a unique split octonion algebra over any field F. When F is algebraically closed or a finite field, these are the only octonion algebras over F. Octonion algebras are always nonassociative. They are, however, alternative algebras, alternativity being a weaker form of associativity. Moreover, the Moufang identitieshold in any octonion algebra. It follows that the invertible elements in any octonion algebra form a Moufang loop, as do the elements of unit norm. The construction of general octonion algebras over an arbitrary field k was described by Leonard Dickson in his book Algebren und ihre Zahlentheorie and repeated by Max Zorn. The product depends on selection of a γ from k. Given q and Q from a quaternion algebra over k, the octonion is written q + Qe. Another octonion may be written r + Re. Then with * denoting the conjugation in the quaternion algebra, their product is Zorn’s German languagedescription of this Cayley–Dickson construction contributed to the persistent use of this eponym describing the construction of composition algebras.
Classification
It is a theorem of Adolf Hurwitz that the F-isomorphism classes of the norm form are in one-to-one correspondence with the isomorphism classes of octonion F-algebras. Moreover, the possible norm forms are exactly the Pfister 3-forms over F. Since any two octonion F-algebras become isomorphic over the algebraic closure of F, one can apply the ideas of non-abelian Galois cohomology. In particular, by using the fact that the automorphism group of the split octonions is the split algebraic group G2, one sees the correspondence of isomorphism classes of octonion F-algebras with isomorphism classes of G2-torsors over F. These isomorphism classes form the non-abelian Galois cohomology set.
