Sedenion

In abstract algebra, sedenions form a 16-dimensional algebra over the reals. The set of sedenions is denoted as $mathbb\{S\}$. Two types are currently known:
# Sedenions obtained by applying the Cayley-Dickson construction
# Conic sedenions ("16-dimensional M-algebra") after Charles Musès, part of his hypernumber concept.

Cayley-Dickson Sedenions

Arithmetic

Like (Cayley-Dickson) octonions, multiplication of Cayley-Dickson 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 power associativity.

Every sedenion is a real linear combination of the unit sedenions 1, "e"₁, "e"₂, "e"₃, "e"₄, "e"₅, "e"₆, "e"₇, "e"₈, "e"₉, "e"₁₀, "e"₁₁, "e"₁₂, "e"₁₃, "e"₁₄ and "e"₁₅,which form a basis of the vector space of sedenions.

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; this means that two non-zero numbers can be multiplied to obtain a zero result: a trivial example is ("e"₃ + "e"₁₀)*("e"₆ - "e"₁₅). All hypercomplex number systems based on the Cayley-Dickson construction from sedenions on contain zero divisors.

The multiplication table of these unit sedenions follows:

Further reading

* Imaeda, K., Imaeda, M.: "Sedenions: algebra and analysis", Applied Mathematics and Computation, 115:77-88 (2000)

* Kinyon, M.K., Phillips, J.D., Vojtěchovský, P.: "C-loops: Extensions and constructions", Journal of Algebra and its Applications 6 (2007), no. 1, 1-20. [http://arxiv.org/abs/math/0412390]

Conic sedenions / "16-dim. M-algebra"

Arithmetic

In contrast to Cayley-Dickson sedenions, which are built on one and 15 roots of negative one, conic sedenions are built on 8 square roots each of positive and negative one. They share non-commutativity and non-associativity with Cayley-Dickson sedenion ("circular sedenion") arithmetic, however, conic sedenions are modular, alternative, and flexible. With the exception of its nilpotents, zero divisors, and zero itself, the arithmetic is closed with respect to the power-of and logarithm operations. Conic sedenions are not power-associative.

For detailed information and isomorphic subalgebras see Musean hypernumber.

ee also

* Hypercomplex number
* Split-complex number