Octonion algebra

In mathematics, an octonion algebra or Cayley algebra over a field F is an algebraic structure which is an 8-dimensional composition algebra over F. In other words, it is a unital nonassociative algebra A over F with a nondegenerate quadratic form N (called the norm form) such that

$N(xy) = N(x)N(y)$

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 octonion algebra for N is a division algebra if and only if the form N is anisotropic. A split octonion algebra is one for which the quadratic form N is isotropic (i.e., there exists a non-zero vector x with N(x) = 0). Up to F-algebra isomorphism, there is a unique split octonion algebra over any field F.^[1] 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 identities hold 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.

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.^[2]

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 $H^1(F, G_2)$.^[3]

See also

• Quaternion algebra

Cryptographic applications

Yongge Wang claimed to have designed efficient fully homomorphic encryption schemes using octonion algebras.^[4]

References

1. Schafer (1995) p.48
2. Lam (2005) p.327
3. Garibaldi, Merkurjev & Serre (2003) pp.9-10,44
4. Yongge Wang. Algebra and Noise-Free Fully Homomorphic Encryption (FHE) Schemes, IACR ePrint, https://eprint.iacr.org/2016/068.pdf Octonion

• Garibaldi, Skip; Merkurjev, Alexander; Serre, Jean-Pierre (2003). Cohomological invariants in Galois cohomology. University Lecture Series 28. Providence, RI: American Mathematical Society. ISBN 0-8218-3287-5. Zbl 1159.12311.
• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.
• Schafer, Richard D. (1995) [1966]. An introduction to non-associative algebras. Dover Publications. ISBN 0-486-68813-5. Zbl 0145.25601.
• Serre, J. P. (2002). Galois Cohomology. Springer Monographs in Mathematics. Translated from the French by Patrick Ion. Berlin: Springer-Verlag. ISBN 3-540-42192-0. Zbl 1004.12003.
• Springer, T. A.; Veldkamp, F. D. (2000). Octonions, Jordan Algebras and Exceptional Groups. Springer-Verlag. ISBN 3-540-66337-1.

External links

• Hazewinkel, Michiel, ed. (2001), "Cayley–Dickson algebra", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4