Sedenion

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. Unlike the octonions, the sedenions are not an alternative algebra. 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 two copies of the biquaternions, or the algebra of 4 by 4 matrices over the reals, or that studied by Smith (1995).

Arithmetic

A visualization of a 4D extension to the cubic Octonion#Fano plane mnemonic,^[1] showing the 35 triads as hyperplanes through the Real (e₀) vertex of the sedenion example given.

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 that, 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

$x=x_{0}e_{0}+x_{1}e_{1}+x_{2}e_{2}+\ldots +x_{{14}}e_{{14}}+x_{{15}}e_{{15}},\,$.

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 they were 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 after sedenions that are based on the Cayley–Dickson construction contain zero divisors.

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:

$e_{0}e_{i}=e_{i}e_{0}=e_{i},$

$e_{i}e_{i}=-e_{0}\,\,{\text{for}}\,\,i\neq 0,$

$e_{i}e_{j}=-e_{j}e_{i}\,\,{\text{for}}\,\,i\neq j\,\,{\text{and}}\,\,i,j\neq 0.$

$e_{i}(e_{j}e_{k})=-(e_{i}e_{j})e_{k}\,\,{\text{for}}\,\,i\neq j,\,\,i,j\neq 0\,\,{\text{and}}\,\,e_{i}e_{j}\neq \pm e_{k}.$

The 35 triads that make up this specific sedenion multiplication table with the 7 triads of the octonion used in creating the sedenion through the Cayley–Dickson construction shown in bold red:

{​{1, 2, 3} , {1, 4, 5} , {1, 7, 6} , {1, 8, 9}, {1, 11, 10}, {1, 13, 12}, {1, 14, 15},
{2, 4, 6} , {2, 5, 7} , {2, 8, 10}, {2, 9, 11}, {2, 14, 12}, {2, 15, 13}, {3, 4, 7} ,
{3, 6, 5} , {3, 8, 11}, {3, 10, 9}, {3, 13, 14}, {3, 15, 12}, {4, 8, 12}, {4, 9, 13},
{4, 10, 14}, {4, 11, 15}, {5, 8, 13}, {5, 10, 15}, {5, 12, 9}, {5, 14, 11}, {6, 8, 14},
{6, 11, 13}, {6, 12, 10}, {6, 15, 9}, {7, 8, 15}, {7, 9, 14}, {7, 12, 11}, {7, 13, 10}​}

The list of 84 sets of zero divisors {e_a, e_b, e_c, e_d}, where (e_a + e_b)$\circ$(e_c + e_d)=0:

Applications

Moreno (1998) showed that the space of pairs of norm-one sedenions that multiply to zero is homeomorphic to the compact form of the exceptional Lie group G₂. (Note that in his paper, a "zero divisor" means a pair of elements that multiply to zero.)

See also

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

Notes

1. (Baez 2002, p. 6)

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
• Baez, John C. (2002). "The Octonions". Bulletin of the American Mathematical Society. 39 (2): 145–205. arXiv:math/0105155v4. doi:10.1090/S0273-0979-01-00934-X. MR 1886087.
• Kinyon, M.K.; Phillips, J.D.; Vojtěchovský, P. (2007). "C-loops: Extensions and constructions". Journal of Algebra and its Applications. 6 (1): 1–20. arXiv:math/0412390. doi:10.1142/S0219498807001990.
• Kivunge, Benard M.; Smith, Jonathan D. H (2004). "Subloops of sedenions" (PDF). Comment. Math. Univ. Carolinae. 45 (2): 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