.

In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the reals obtained by applying the Cayley–Dickson construction to the octonions. The set of sedenions is denoted by \( \mathbb{S} \) .

The term "sedenion" is also used for other 16-dimensional algebraic structures, such as a tensor product of 2 copies of the biquaternions, or the algebra of 4 by 4 matrices over the reals, or that studied by Smith (1995).

Arithmetic

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 power associativity, which can be stated as for any element x of \( \mathbb{S} \) , the power \( x^n \) is well-defined. They are also flexible.

Every sedenion is a linear combination of the unit sedenions e₀, e₁, e₂, e₃, ...,e₁₅, which form a basis of the vector space of sedenions. Every sedenion can be represented in the form

\( x = x_0e_0 + x_1e_1 + x_2e_2 + \ldots + x_{14}e_{14} + x_{15}e_{15},\,. \)

Addition and subtraction are defined by the addition and subtraction of corresponding coefficients and multiplication is distributive over addition.

Like other algebras based on the Cayley–Dickson construction, the sedenions contain the algebra it was constructed from. So they contain the octonions (e₀ to e₇ in the table below), and therefore also the quaternions (e₀ to e₃), complex numbers (e₀ and e₁) and reals (e₀).

The sedenions have a multiplicative identity element e₀ and multiplicative inverses but they are not a division algebra because they have zero divisors. This means that two non-zero sedenions can be multiplied to obtain zero: an example is (e₃ + e₁₀)×(e₆ − e₁₅). All hypercomplex number systems based on the Cayley–Dickson construction after sedenions contain zero divisors.

The multiplication table of these unit sedenions follows:

The multiplication table of these unit sedenions follows:

×	e0	e1	e2	e3	e4	e5	e6	e7	e8	e9	e10	e11	e12	e13	e14	e15
e0	e0	e1	e2	e3	e4	e5	e6	e7	e8	e9	e10	e11	e12	e13	e14	e15
e1	e1	−e0	e3	−e2	e5	−e4	−e7	e6	e9	−e8	−e11	e10	−e13	e12	e15	−e14
e2	e2	−e3	−e0	e1	e6	e7	−e4	−e5	e10	e11	−e8	−e9	−e14	−e15	e12	e13
e3	e3	e2	−e1	−e0	e7	−e6	e5	−e4	e11	−e10	e9	−e8	−e15	e14	−e13	e12
e4	e4	−e5	−e6	−e7	−e0	e1	e2	e3	e12	e13	e14	e15	−e8	−e9	−e10	−e11
e5	e5	e4	−e7	e6	−e1	−e0	−e3	e2	e13	−e12	e15	−e14	e9	−e8	e11	−e10
e6	e6	e7	e4	−e5	−e2	e3	−e0	−e1	e14	−e15	−e12	e13	e10	−e11	−e8	e9
e7	e7	−e6	e5	e4	−e3	−e2	e1	−e0	e15	e14	−e13	−e12	e11	e10	−e9	−e8
e8	e8	−e9	−e10	−e11	−e12	−e13	−e14	−e15	−e0	e1	e2	e3	e4	e5	e6	e7
e9	e9	e8	−e11	e10	−e13	e12	e15	−e14	−e1	−e0	−e3	e2	−e5	e4	e7	−e6
e10	e10	e11	e8	−e9	−e14	−e15	e12	e13	−e2	e3	−e0	−e1	−e6	−e7	e4	e5
e11	e11	−e10	e9	e8	−e15	e14	−e13	e12	−e3	−e2	e1	−e0	−e7	e6	−e5	e4
e12	e12	e13	e14	e15	e8	−e9	−e10	−e11	−e4	e5	e6	e7	−e0	−e1	−e2	−e3
e13	e13	−e12	e15	−e14	e9	e8	e11	−e10	−e5	−e4	e7	−e6	e1	−e0	e3	−e2
e14	e14	−e15	−e12	e13	e10	−e11	e8	e9	−e6	−e7	−e4	e5	e2	−e3	−e0	e1
e15	e15	e14	−e13	−e12	e11	e10	−e9	e8	−e7	e6	−e5	−e4	e3	e2	−e1	−e0

From the above table, we can see that:

\( e_0e_i = e_ie_0 = e_i, \)

\( e_ie_i = -e_0 \,\, \text{for}\,\, i \neq 0, \)

\( e_ie_j = -e_je_i \,\, \text{for}\,\, i \neq j \,\,\text{and}\,\, i,j \neq 0. \)

\( e_i(e_je_k) = -(e_ie_j)e_k \,\,\text{for}\,\, i \neq j,\,\, i,j \neq 0\,\,\text{and}\,\, e_ie_j\neq \plusmn e_k. \)

Applications

Moreno (1998) showed that the space of norm 1 zero-divisors of the sedenions is homeomorphic to the compact form of the exceptional Lie group G₂.

See also

Pfister's sixteen-square identity
Hypercomplex number
Split-complex number

References

Imaeda, K.; Imaeda, M. (2000), "Sedenions: algebra and analysis", Applied mathematics and computation 115 (2): 77–88, doi:10.1016/S0096-3003(99)00140-X, MR 1786945
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. [1]
Kivunge, Benard M. and Smith, Jonathan D. H: "Subloops of sedenions", Comment.Math.Univ.Carolinae 45,2 (2004)295–302.
Moreno, Guillermo (1998), "The zero divisors of the Cayley–Dickson algebras over the real numbers", Sociedad Matemática Mexicana. Boletí n. Tercera Serie 4 (1): 13–28, arXiv:q-alg/9710013, MR 1625585
Smith, Jonathan D. H. (1995), "A left loop on the 15-sphere", Journal of Algebra 176 (1): 128–138, doi:10.1006/jabr.1995.1237, MR 1345298

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