Sedenion

The sedenions form a 16-dimensional algebra over the reals obtained by applying the Cayley-Dickson construction to the octonions.

Like octonions, multiplication of sedenions is neither commutative nor associative. But in contrast to the octonions, the sedenions do not even have the property of being alternative. They do, however, have the property of being power-associative.

The sedenions have a multiplicative identity element 1 and multiplicative inverses, but they are not a division algebra. This is because they have zero divisors.

Every sedenion is a real linear combination of the unit sedenions 1, e₁, e₂, e₃, e₄, e₅, e₆, e₇, e₈, e₉, e₁₀, e₁₁, e₁₂, e₁₃, e₁₄ and e₁₅, which form a basis of the vector space of sedenions. The multiplication table of these unit sedenions looks as follows.

Further reading

• Carmody, Kevin: Circular and Hyperbolic Quaternions, Octonions and Sedenions, Applied Mathematics and Computation 28:47-72 (1988)
• Carmody, Kevin: Circular and Hyperbolic Quaternions, Octonions and Sedenions - Further results, Applied Mathematics and Computation, 84:27-47 (1997)
• Imaeda, K., Imaeda, M.: Sedenions: algebra and analysis, Applied Mathematics and Computation, 115:77-88 (2000)