Sedenion

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The Sedenions (symbol ) are 16-dimensional hyper-complex numbers . They arise from the application of the doubling process from the octonions . ${\displaystyle \mathbb {S} }$

The multiplication of the sedions is neither commutative nor alternative (and therefore not associative ). It is only potency-associative and flexible . Furthermore, the sedions fulfill the Jordan identity and therefore form a non-commutative Jordan algebra . Sedenions have zero divisors .

Each Sedenion is a real linear combination of the units , where : ${\displaystyle e_{i},\ i=0,\ldots ,15}$ ${\displaystyle e_{0}=1}$

${\displaystyle s=\sum _{i=0}^{15}x_{i}e_{i}=x_{0}+x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}+x_{4}e_{4}+x_{5}e_{5}+x_{6}e_{6}+x_{7}e_{7}+x_{8}e_{8}+x_{9}e_{9}+x_{10}e_{10}+x_{11}e_{11}+x_{12}e_{12}+x_{13}e_{13}+x_{14}e_{14}+x_{15}e_{15}}$

multiplication

A possible multiplication table for the units is:

⋅	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	e 15
1	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	e 15
e 1	e 1	−1	e 3	−e 2	e 5	−e 4	−e 7	e 6	e 9	−e 8	−e 11	e 10	−e 13	e 12	e 15	−e 14
e 2	e 2	−e 3	−1	e 1	e 6	e 7	−e 4	−e 5	e 10	e 11	−e 8	−e 9	−e 14	−e 15	e 12	e 13
e 3	e 3	e 2	−e 1	−1	e 7	−e 6	e 5	−e 4	e 11	−e 10	e 9	−e 8	−e 15	e 14	−e 13	e 12
e 4	e 4	−e 5	−e 6	−e 7	−1	e 1	e 2	e 3	e 12	e 13	e 14	e 15	−e 8	−e 9	−e 10	−e 11
e 5	e 5	e 4	−e 7	e 6	−e 1	−1	−e 3	e 2	e 13	−e 12	e 15	−e 14	e 9	−e 8	e 11	−e 10
e 6	e 6	e 7	e 4	−e 5	−e 2	e 3	−1	−e 1	e 14	−e 15	−e 12	e 13	e 10	−e 11	−e 8	e 9
e 7	e 7	−e 6	e 5	e 4	−e 3	−e 2	e 1	−1	e 15	e 14	−e 13	−e 12	e 11	e 10	−e 9	−e 8
e 8	e 8	−e 9	−e 10	−e 11	−e 12	−e 13	−e 14	−e 15	−1	e 1	e 2	e 3	e 4	e 5	e 6	e 7
e 9	e 9	e 8	−e 11	e 10	−e 13	e 12	e 15	−e 14	−e 1	−1	−e 3	e 2	−e 5	e 4	e 7	−e 6
e 10	e 10	e 11	e 8	−e 9	−e 14	−e 15	e 12	e 13	−e 2	e 3	−1	−e 1	−e 6	−e 7	e 4	e 5
e 11	e 11	−e 10	e 9	e 8	−e 15	e 14	−e 13	e 12	−e 3	−e 2	e 1	−1	−e 7	e 6	−e 5	e 4
e 12	e 12	e 13	e 14	e 15	e 8	−e 9	−e 10	−e 11	−e 4	e 5	e 6	e 7	−1	−e 1	−e 2	−e 3
e 13	e 13	−e 12	e 15	−e 14	e 9	e 8	e 11	−e 10	−e 5	−e 4	e 7	−e 6	e 1	−1	e 3	−e 2
e 14	e 14	−e 15	−e 12	e 13	e 10	−e 11	e 8	e 9	−e 6	−e 7	−e 4	e 5	e 2	−e 3	−1	e 1
e 15	e 15	e 14	−e 13	−e 12	e 11	e 10	−e 9	e 8	−e 7	e 6	−e 5	−e 4	e 3	e 2	−e 1	−1

The left column is to be read as the first or left factor, the top line as the second or right factor:, but see also anti-commutativity . ${\displaystyle e_{1}\cdot e_{2}=e_{3}}$ ${\displaystyle e_{2}\cdot e_{1}=-e_{3}}$

It applies

${\displaystyle \prod _{i=0}^{15}e_{i}=e_{0}e_{1}e_{2}\cdots e_{15}=-1.}$

Zero divisor

The table shows that the sedions have zero divisors:

${\displaystyle (e_{3}+e_{10})\cdot (e_{6}-e_{15})=e_{3}e_{6}+e_{10}e_{6}-e_{3}e_{15}-e_{10}e_{15}=e_{5}+e_{12}-e_{12}-e_{5}=0}$

The space of the zero divisors with norm 1 is homeomorphic to the compact form of the exceptional Lie group G ₂ .

Individual evidence

1. ^ R. Guillermo Moreno (1997): The zero divisors of the Cayley-Dickson algebras over the real numbers , arxiv : q-alg / 9710013 .