Hypercomplex number system
In abstract algebra , the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the real numbers , usually represented by the capital letter S, boldface S blackboard bold
S
{\displaystyle \mathbb {S} }
Cayley–Dickson construction to the octonions , and as such the octonions are isomorphic to a subalgebra of the sedenions. Unlike the octonions, the sedenions are not an alternative algebra . Applying the Cayley–Dickson construction to the sedenions yields a 32-dimensional algebra, sometimes called the 32-ions or trigintaduonions .[1]
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 × 4 matrices over the real numbers, or that studied by Smith (1995) .
A visualization of a 4D extension to the cubic octonion ,[2] hyperplanes through the real
(
e
0
)
{\displaystyle (e_{0})}
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
S
{\displaystyle \mathbb {S} }
x
n
{\displaystyle x^{n}}
flexible .
Every sedenion is a linear combination of the unit sedenions
e
0
{\displaystyle e_{0}}
e
1
{\displaystyle e_{1}}
e
2
{\displaystyle e_{2}}
e
3
{\displaystyle e_{3}}
e
15
{\displaystyle e_{15}}
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
+
⋯
+
x
14
e
14
+
x
15
e
15
.
{\displaystyle x=x_{0}e_{0}+x_{1}e_{1}+x_{2}e_{2}+\cdots +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 (generated by
e
0
{\displaystyle e_{0}}
e
7
{\displaystyle e_{7}}
quaternions (generated by
e
0
{\displaystyle e_{0}}
e
3
{\displaystyle e_{3}}
complex numbers (generated by
e
0
{\displaystyle e_{0}}
e
1
{\displaystyle e_{1}}
e
0
{\displaystyle e_{0}}
The sedenions have a multiplicative identity element
e
0
{\displaystyle e_{0}}
division algebra because they have zero divisors . This means that two nonzero sedenions can be multiplied to obtain zero: an example is
(
e
3
+
e
10
)
(
e
6
−
e
15
)
{\displaystyle (e_{3}+e_{10})(e_{6}-e_{15})}
hypercomplex number systems after sedenions that are based on the Cayley–Dickson construction also contain zero divisors.
A sedenion multiplication table is shown below:
e
i
e
j
{\displaystyle e_{i}e_{j}}
e
j
{\displaystyle e_{j}}
e
0
{\displaystyle e_{0}}
e
1
{\displaystyle e_{1}}
e
2
{\displaystyle e_{2}}
e
3
{\displaystyle e_{3}}
e
4
{\displaystyle e_{4}}
e
5
{\displaystyle e_{5}}
e
6
{\displaystyle e_{6}}
e
7
{\displaystyle e_{7}}
e
8
{\displaystyle e_{8}}
e
9
{\displaystyle e_{9}}
e
10
{\displaystyle e_{10}}
e
11
{\displaystyle e_{11}}
e
12
{\displaystyle e_{12}}
e
13
{\displaystyle e_{13}}
e
14
{\displaystyle e_{14}}
e
15
{\displaystyle e_{15}}
e
i
{\displaystyle e_{i}}
e
0
{\displaystyle e_{0}}
e
0
{\displaystyle e_{0}}
e
1
{\displaystyle e_{1}}
e
2
{\displaystyle e_{2}}
e
3
{\displaystyle e_{3}}
e
4
{\displaystyle e_{4}}
e
5
{\displaystyle e_{5}}
e
6
{\displaystyle e_{6}}
e
7
{\displaystyle e_{7}}
e
8
{\displaystyle e_{8}}
e
9
{\displaystyle e_{9}}
e
10
{\displaystyle e_{10}}
e
11
{\displaystyle e_{11}}
e
12
{\displaystyle e_{12}}
e
13
{\displaystyle e_{13}}
e
14
{\displaystyle e_{14}}
e
15
{\displaystyle e_{15}}
e
1
{\displaystyle e_{1}}
e
1
{\displaystyle e_{1}}
−
e
0
{\displaystyle -e_{0}}
e
3
{\displaystyle e_{3}}
−
e
2
{\displaystyle -e_{2}}
e
5
{\displaystyle e_{5}}
−
e
4
{\displaystyle -e_{4}}
−
e
7
{\displaystyle -e_{7}}
e
6
{\displaystyle e_{6}}
e
9
{\displaystyle e_{9}}
−
e
8
{\displaystyle -e_{8}}
−
e
11
{\displaystyle -e_{11}}
e
10
{\displaystyle e_{10}}
−
e
13
{\displaystyle -e_{13}}
e
12
{\displaystyle e_{12}}
e
15
{\displaystyle e_{15}}
−
e
14
{\displaystyle -e_{14}}
e
2
{\displaystyle e_{2}}
e
2
{\displaystyle e_{2}}
−
e
3
{\displaystyle -e_{3}}
−
e
0
{\displaystyle -e_{0}}
e
1
{\displaystyle e_{1}}
e
6
{\displaystyle e_{6}}
e
7
{\displaystyle e_{7}}
−
e
4
{\displaystyle -e_{4}}
−
e
5
{\displaystyle -e_{5}}
e
10
{\displaystyle e_{10}}
e
11
{\displaystyle e_{11}}
−
e
8
{\displaystyle -e_{8}}
−
e
9
{\displaystyle -e_{9}}
−
e
14
{\displaystyle -e_{14}}
−
e
15
{\displaystyle -e_{15}}
e
12
{\displaystyle e_{12}}
e
13
{\displaystyle e_{13}}
e
3
{\displaystyle e_{3}}
e
3
{\displaystyle e_{3}}
e
2
{\displaystyle e_{2}}
−
e
1
{\displaystyle -e_{1}}
−
e
0
{\displaystyle -e_{0}}
e
7
{\displaystyle e_{7}}
−
e
6
{\displaystyle -e_{6}}
e
5
{\displaystyle e_{5}}
−
e
4
{\displaystyle -e_{4}}
e
11
{\displaystyle e_{11}}
−
e
10
{\displaystyle -e_{10}}
e
9
{\displaystyle e_{9}}
−
e
8
{\displaystyle -e_{8}}
−
e
15
{\displaystyle -e_{15}}
e
14
{\displaystyle e_{14}}
−
e
13
{\displaystyle -e_{13}}
e
12
{\displaystyle e_{12}}
e
4
{\displaystyle e_{4}}
e
4
{\displaystyle e_{4}}
−
e
5
{\displaystyle -e_{5}}
−
e
6
{\displaystyle -e_{6}}
−
e
7
{\displaystyle -e_{7}}
−
e
0
{\displaystyle -e_{0}}
e
1
{\displaystyle e_{1}}
e
2
{\displaystyle e_{2}}
e
3
{\displaystyle e_{3}}
e
12
{\displaystyle e_{12}}
e
13
{\displaystyle e_{13}}
e
14
{\displaystyle e_{14}}
e
15
{\displaystyle e_{15}}
−
e
8
{\displaystyle -e_{8}}
−
e
9
{\displaystyle -e_{9}}
−
e
10
{\displaystyle -e_{10}}
−
e
11
{\displaystyle -e_{11}}
e
5
{\displaystyle e_{5}}
e
5
{\displaystyle e_{5}}
e
4
{\displaystyle e_{4}}
−
e
7
{\displaystyle -e_{7}}
e
6
{\displaystyle e_{6}}
−
e
1
{\displaystyle -e_{1}}
−
e
0
{\displaystyle -e_{0}}
−
e
3
{\displaystyle -e_{3}}
e
2
{\displaystyle e_{2}}
e
13
{\displaystyle e_{13}}
−
e
12
{\displaystyle -e_{12}}
e
15
{\displaystyle e_{15}}
−
e
14
{\displaystyle -e_{14}}
e
9
{\displaystyle e_{9}}
−
e
8
{\displaystyle -e_{8}}
e
11
{\displaystyle e_{11}}
−
e
10
{\displaystyle -e_{10}}
e
6
{\displaystyle e_{6}}
e
6
{\displaystyle e_{6}}
e
7
{\displaystyle e_{7}}
e
4
{\displaystyle e_{4}}
−
e
5
{\displaystyle -e_{5}}
−
e
2
{\displaystyle -e_{2}}
e
3
{\displaystyle e_{3}}
−
e
0
{\displaystyle -e_{0}}
−
e
1
{\displaystyle -e_{1}}
e
14
{\displaystyle e_{14}}
−
e
15
{\displaystyle -e_{15}}
−
e
12
{\displaystyle -e_{12}}
e
13
{\displaystyle e_{13}}
e
10
{\displaystyle e_{10}}
−
e
11
{\displaystyle -e_{11}}
−
e
8
{\displaystyle -e_{8}}
e
9
{\displaystyle e_{9}}
e
7
{\displaystyle e_{7}}
e
7
{\displaystyle e_{7}}
−
e
6
{\displaystyle -e_{6}}
e
5
{\displaystyle e_{5}}
e
4
{\displaystyle e_{4}}
−
e
3
{\displaystyle -e_{3}}
−
e
2
{\displaystyle -e_{2}}
e
1
{\displaystyle e_{1}}
−
e
0
{\displaystyle -e_{0}}
e
15
{\displaystyle e_{15}}
e
14
{\displaystyle e_{14}}
−
e
13
{\displaystyle -e_{13}}
−
e
12
{\displaystyle -e_{12}}
e
11
{\displaystyle e_{11}}
e
10
{\displaystyle e_{10}}
−
e
9
{\displaystyle -e_{9}}
−
e
8
{\displaystyle -e_{8}}
e
8
{\displaystyle e_{8}}
e
8
{\displaystyle e_{8}}
−
e
9
{\displaystyle -e_{9}}
−
e
10
{\displaystyle -e_{10}}
−
e
11
{\displaystyle -e_{11}}
−
e
12
{\displaystyle -e_{12}}
−
e
13
{\displaystyle -e_{13}}
−
e
14
{\displaystyle -e_{14}}
−
e
15
{\displaystyle -e_{15}}
−
e
0
{\displaystyle -e_{0}}
e
1
{\displaystyle e_{1}}
e
2
{\displaystyle e_{2}}
e
3
{\displaystyle e_{3}}
e
4
{\displaystyle e_{4}}
e
5
{\displaystyle e_{5}}
e
6
{\displaystyle e_{6}}
e
7
{\displaystyle e_{7}}
e
9
{\displaystyle e_{9}}
e
9
{\displaystyle e_{9}}
e
8
{\displaystyle e_{8}}
−
e
11
{\displaystyle -e_{11}}
e
10
{\displaystyle e_{10}}
−
e
13
{\displaystyle -e_{13}}
e
12
{\displaystyle e_{12}}
e
15
{\displaystyle e_{15}}
−
e
14
{\displaystyle -e_{14}}
−
e
1
{\displaystyle -e_{1}}
−
e
0
{\displaystyle -e_{0}}
−
e
3
{\displaystyle -e_{3}}
e
2
{\displaystyle e_{2}}
−
e
5
{\displaystyle -e_{5}}
e
4
{\displaystyle e_{4}}
e
7
{\displaystyle e_{7}}
−
e
6
{\displaystyle -e_{6}}
e
10
{\displaystyle e_{10}}
e
10
{\displaystyle e_{10}}
e
11
{\displaystyle e_{11}}
e
8
{\displaystyle e_{8}}
−
e
9
{\displaystyle -e_{9}}
−
e
14
{\displaystyle -e_{14}}
−
e
15
{\displaystyle -e_{15}}
e
12
{\displaystyle e_{12}}
e
13
{\displaystyle e_{13}}
−
e
2
{\displaystyle -e_{2}}
e
3
{\displaystyle e_{3}}
−
e
0
{\displaystyle -e_{0}}
−
e
1
{\displaystyle -e_{1}}
−
e
6
{\displaystyle -e_{6}}
−
e
7
{\displaystyle -e_{7}}
e
4
{\displaystyle e_{4}}
e
5
{\displaystyle e_{5}}
e
11
{\displaystyle e_{11}}
e
11
{\displaystyle e_{11}}
−
e
10
{\displaystyle -e_{10}}
e
9
{\displaystyle e_{9}}
e
8
{\displaystyle e_{8}}
−
e
15
{\displaystyle -e_{15}}
e
14
{\displaystyle e_{14}}
−
e
13
{\displaystyle -e_{13}}
e
12
{\displaystyle e_{12}}
−
e
3
{\displaystyle -e_{3}}
−
e
2
{\displaystyle -e_{2}}
e
1
{\displaystyle e_{1}}
−
e
0
{\displaystyle -e_{0}}
−
e
7
{\displaystyle -e_{7}}
e
6
{\displaystyle e_{6}}
−
e
5
{\displaystyle -e_{5}}
e
4
{\displaystyle e_{4}}
e
12
{\displaystyle e_{12}}
e
12
{\displaystyle e_{12}}
e
13
{\displaystyle e_{13}}
e
14
{\displaystyle e_{14}}
e
15
{\displaystyle e_{15}}
e
8
{\displaystyle e_{8}}
−
e
9
{\displaystyle -e_{9}}
−
e
10
{\displaystyle -e_{10}}
−
e
11
{\displaystyle -e_{11}}
−
e
4
{\displaystyle -e_{4}}
e
5
{\displaystyle e_{5}}
e
6
{\displaystyle e_{6}}
e
7
{\displaystyle e_{7}}
−
e
0
{\displaystyle -e_{0}}
−
e
1
{\displaystyle -e_{1}}
−
e
2
{\displaystyle -e_{2}}
−
e
3
{\displaystyle -e_{3}}
e
13
{\displaystyle e_{13}}
e
13
{\displaystyle e_{13}}
−
e
12
{\displaystyle -e_{12}}
e
15
{\displaystyle e_{15}}
−
e
14
{\displaystyle -e_{14}}
e
9
{\displaystyle e_{9}}
e
8
{\displaystyle e_{8}}
e
11
{\displaystyle e_{11}}
−
e
10
{\displaystyle -e_{10}}
−
e
5
{\displaystyle -e_{5}}
−
e
4
{\displaystyle -e_{4}}
e
7
{\displaystyle e_{7}}
−
e
6
{\displaystyle -e_{6}}
e
1
{\displaystyle e_{1}}
−
e
0
{\displaystyle -e_{0}}
e
3
{\displaystyle e_{3}}
−
e
2
{\displaystyle -e_{2}}
e
14
{\displaystyle e_{14}}
e
14
{\displaystyle e_{14}}
−
e
15
{\displaystyle -e_{15}}
−
e
12
{\displaystyle -e_{12}}
e
13
{\displaystyle e_{13}}
e
10
{\displaystyle e_{10}}
−
e
11
{\displaystyle -e_{11}}
e
8
{\displaystyle e_{8}}
e
9
{\displaystyle e_{9}}
−
e
6
{\displaystyle -e_{6}}
−
e
7
{\displaystyle -e_{7}}
−
e
4
{\displaystyle -e_{4}}
e
5
{\displaystyle e_{5}}
e
2
{\displaystyle e_{2}}
−
e
3
{\displaystyle -e_{3}}
−
e
0
{\displaystyle -e_{0}}
e
1
{\displaystyle e_{1}}
e
15
{\displaystyle e_{15}}
e
15
{\displaystyle e_{15}}
e
14
{\displaystyle e_{14}}
−
e
13
{\displaystyle -e_{13}}
−
e
12
{\displaystyle -e_{12}}
e
11
{\displaystyle e_{11}}
e
10
{\displaystyle e_{10}}
−
e
9
{\displaystyle -e_{9}}
e
8
{\displaystyle e_{8}}
−
e
7
{\displaystyle -e_{7}}
e
6
{\displaystyle e_{6}}
−
e
5
{\displaystyle -e_{5}}
−
e
4
{\displaystyle -e_{4}}
e
3
{\displaystyle e_{3}}
e
2
{\displaystyle e_{2}}
−
e
1
{\displaystyle -e_{1}}
−
e
0
{\displaystyle -e_{0}}
Sedenion properties [ edit ] From the above table, we can see that:
e
0
e
i
=
e
i
e
0
=
e
i
for all
i
,
{\displaystyle e_{0}e_{i}=e_{i}e_{0}=e_{i}\,{\text{for all}}\,i,}
e
i
e
i
=
−
e
0
for
i
≠
0
,
{\displaystyle e_{i}e_{i}=-e_{0}\,\,{\text{for}}\,\,i\neq 0,}
e
i
e
j
=
−
e
j
e
i
for
i
≠
j
with
i
,
j
≠
0.
{\displaystyle e_{i}e_{j}=-e_{j}e_{i}\,\,{\text{for}}\,\,i\neq j\,\,{\text{with}}\,\,i,j\neq 0.}
The sedenions are not fully anti-associative. Choose any four generators,
i
,
j
,
k
{\displaystyle i,j,k}
l
{\displaystyle l}
(
i
j
)
(
k
l
)
=
−
(
(
i
j
)
k
)
l
=
(
i
(
j
k
)
)
l
=
−
i
(
(
j
k
)
l
)
=
i
(
j
(
k
l
)
)
=
−
(
i
j
)
(
k
l
)
=
0
{\displaystyle (ij)(kl)=-((ij)k)l=(i(jk))l=-i((jk)l)=i(j(kl))=-(ij)(kl)=0}
In particular, in the table above, using
e
1
,
e
2
,
e
4
{\displaystyle e_{1},e_{2},e_{4}}
e
8
{\displaystyle e_{8}}
(
e
1
e
2
)
e
12
=
e
1
(
e
2
e
12
)
=
−
e
15
{\displaystyle (e_{1}e_{2})e_{12}=e_{1}(e_{2}e_{12})=-e_{15}}
Quaternionic subalgebras [ edit ] The 35 triads that make up this specific sedenion multiplication table with the 7 triads of the octonions used in creating the sedenion through the Cayley–Dickson construction shown in bold:
The binary representations of the indices of these triples bitwise XOR to 0.
{ {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},
The list of 84 sets of zero divisors
{
e
a
,
e
b
,
e
c
,
e
d
}
{\displaystyle \{e_{a},e_{b},e_{c},e_{d}\}}
(
e
a
+
e
b
)
∘
(
e
c
+
e
d
)
=
0
{\displaystyle (e_{a}+e_{b})\circ (e_{c}+e_{d})=0}
Sedenion Zero Divisors
{
e
a
,
e
b
,
e
c
,
e
d
}
where
(
e
a
+
e
b
)
∘
(
e
c
+
e
d
)
=
0
1
≤
a
≤
6
,
c
>
a
,
9
≤
b
≤
15
9
≤
c
≤
15
−
9
≥
d
≥
−
15
{
e
1
,
e
10
,
e
5
,
e
14
}
{
e
1
,
e
10
,
e
4
,
−
e
15
}
{
e
1
,
e
10
,
e
7
,
e
12
}
{
e
1
,
e
10
,
e
6
,
−
e
13
}
{
e
1
,
e
11
,
e
4
,
e
14
}
{
e
1
,
e
11
,
e
6
,
−
e
12
}
{
e
1
,
e
11
,
e
5
,
e
15
}
{
e
1
,
e
11
,
e
7
,
−
e
13
}
{
e
1
,
e
12
,
e
2
,
e
15
}
{
e
1
,
e
12
,
e
3
,
−
e
14
}
{
e
1
,
e
12
,
e
6
,
e
11
}
{
e
1
,
e
12
,
e
7
,
−
e
10
}
{
e
1
,
e
13
,
e
6
,
e
10
}
{
e
1
,
e
13
,
e
2
,
−
e
14
}
{
e
1
,
e
13
,
e
7
,
e
11
}
{
e
1
,
e
13
,
e
3
,
−
e
15
}
{
e
1
,
e
14
,
e
2
,
e
13
}
{
e
1
,
e
14
,
e
4
,
−
e
11
}
{
e
1
,
e
14
,
e
3
,
e
12
}
{
e
1
,
e
14
,
e
5
,
−
e
10
}
{
e
1
,
e
15
,
e
3
,
e
13
}
{
e
1
,
e
15
,
e
2
,
−
e
12
}
{
e
1
,
e
15
,
e
4
,
e
10
}
{
e
1
,
e
15
,
e
5
,
−
e
11
}
{
e
2
,
e
9
,
e
4
,
e
15
}
{
e
2
,
e
9
,
e
5
,
−
e
14
}
{
e
2
,
e
9
,
e
6
,
e
13
}
{
e
2
,
e
9
,
e
7
,
−
e
12
}
{
e
2
,
e
11
,
e
5
,
e
12
}
{
e
2
,
e
11
,
e
4
,
−
e
13
}
{
e
2
,
e
11
,
e
6
,
e
15
}
{
e
2
,
e
11
,
e
7
,
−
e
14
}
{
e
2
,
e
12
,
e
3
,
e
13
}
{
e
2
,
e
12
,
e
5
,
−
e
11
}
{
e
2
,
e
12
,
e
7
,
e
9
}
{
e
2
,
e
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{\displaystyle {\begin{array}{c}{\text{Sedenion Zero Divisors}}\quad \{e_{a},e_{b},e_{c},e_{d}\}\\{\text{where}}~(e_{a}+e_{b})\circ (e_{c}+e_{d})=0\\{\begin{array}{ccc}1\leq a\leq 6,&c>a,&9\leq b\leq 15\\9\leq c\leq 15&&-9\geq d\geq -15\end{array}}\\{\begin{array}{ll}\{e_{1},e_{10},e_{5},e_{14}\}&\{e_{1},e_{10},e_{4},-e_{15}\}\\\{e_{1},e_{10},e_{7},e_{12}\}&\{e_{1},e_{10},e_{6},-e_{13}\}\\\{e_{1},e_{11},e_{4},e_{14}\}&\{e_{1},e_{11},e_{6},-e_{12}\}\\\{e_{1},e_{11},e_{5},e_{15}\}&\{e_{1},e_{11},e_{7},-e_{13}\}\\\{e_{1},e_{12},e_{2},e_{15}\}&\{e_{1},e_{12},e_{3},-e_{14}\}\\\{e_{1},e_{12},e_{6},e_{11}\}&\{e_{1},e_{12},e_{7},-e_{10}\}\\\{e_{1},e_{13},e_{6},e_{10}\}&\{e_{1},e_{13},e_{2},-e_{14}\}\\\{e_{1},e_{13},e_{7},e_{11}\}&\{e_{1},e_{13},e_{3},-e_{15}\}\\\{e_{1},e_{14},e_{2},e_{13}\}&\{e_{1},e_{14},e_{4},-e_{11}\}\\\{e_{1},e_{14},e_{3},e_{12}\}&\{e_{1},e_{14},e_{5},-e_{10}\}\\\{e_{1},e_{15},e_{3},e_{13}\}&\{e_{1},e_{15},e_{2},-e_{12}\}\\\{e_{1},e_{15},e_{4},e_{10}\}&\{e_{1},e_{15},e_{5},-e_{11}\}\\\{e_{2},e_{9},e_{4},e_{15}\}&\{e_{2},e_{9},e_{5},-e_{14}\}\\\{e_{2},e_{9},e_{6},e_{13}\}&\{e_{2},e_{9},e_{7},-e_{12}\}\\\{e_{2},e_{11},e_{5},e_{12}\}&\{e_{2},e_{11},e_{4},-e_{13}\}\\\{e_{2},e_{11},e_{6},e_{15}\}&\{e_{2},e_{11},e_{7},-e_{14}\}\\\{e_{2},e_{12},e_{3},e_{13}\}&\{e_{2},e_{12},e_{5},-e_{11}\}\\\{e_{2},e_{12},e_{7},e_{9}\}&\{e_{2},e_{13},e_{3},-e_{12}\}\\\{e_{2},e_{13},e_{4},e_{11}\}&\{e_{2},e_{13},e_{6},-e_{9}\}\\\{e_{2},e_{14},e_{5},e_{9}\}&\{e_{2},e_{14},e_{3},-e_{15}\}\\\{e_{2},e_{14},e_{3},e_{14}\}&\{e_{2},e_{15},e_{4},-e_{9}\}\\\{e_{2},e_{15},e_{3},e_{14}\}&\{e_{2},e_{15},e_{6},-e_{11}\}\\\{e_{3},e_{9},e_{6},e_{12}\}&\{e_{3},e_{9},e_{4},-e_{14}\}\\\{e_{3},e_{9},e_{7},e_{13}\}&\{e_{3},e_{9},e_{5},-e_{15}\}\\\{e_{3},e_{10},e_{4},e_{13}\}&\{e_{3},e_{10},e_{5},-e_{12}\}\\\{e_{3},e_{10},e_{7},e_{14}\}&\{e_{3},e_{10},e_{6},-e_{15}\}\\\{e_{3},e_{12},e_{5},e_{10}\}&\{e_{3},e_{12},e_{6},-e_{9}\}\\\{e_{3},e_{14},e_{4},e_{9}\}&\{e_{3},e_{13},e_{4},-e_{10}\}\\\{e_{3},e_{15},e_{5},e_{9}\}&\{e_{3},e_{13},e_{7},-e_{9}\}\\\{e_{3},e_{15},e_{6},e_{10}\}&\{e_{3},e_{14},e_{7},-e_{10}\}\\\{e_{4},e_{9},e_{7},e_{10}\}&\{e_{4},e_{9},e_{6},-e_{11}\}\\\{e_{4},e_{10},e_{5},e_{11}\}&\{e_{4},e_{10},e_{7},-e_{9}\}\\\{e_{4},e_{11},e_{6},e_{9}\}&\{e_{4},e_{11},e_{5},-e_{10}\}\\\{e_{4},e_{13},e_{6},e_{15}\}&\{e_{4},e_{13},e_{7},-e_{14}\}\\\{e_{4},e_{14},e_{7},e_{13}\}&\{e_{4},e_{14},e_{5},-e_{15}\}\\\{e_{4},e_{15},e_{5},e_{14}\}&\{e_{4},e_{15},e_{6},-e_{13}\}\\\{e_{5},e_{10},e_{6},e_{9}\}&\{e_{5},e_{9},e_{6},-e_{10}\}\\\{e_{5},e_{11},e_{7},e_{9}\}&\{e_{5},e_{9},e_{7},-e_{11}\}\\\{e_{5},e_{12},e_{7},e_{14}\}&\{e_{5},e_{12},e_{6},-e_{15}\}\\\{e_{5},e_{15},e_{6},e_{12}\}&\{e_{5},e_{14},e_{7},-e_{12}\}\\\{e_{6},e_{11},e_{7},e_{10}\}&\{e_{6},e_{10},e_{7},-e_{11}\}\\\{e_{6},e_{13},e_{7},e_{12}\}&\{e_{6},e_{12},e_{7},-e_{13}\}\end{array}}\end{array}}}
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.)
Guillard & Gresnigt (2019) demonstrated that the three generations of leptons and quarks that are associated with unbroken gauge symmetry
S
U
(
3
)
c
×
U
(
1
)
e
m
{\displaystyle \mathrm {SU(3)_{c}\times U(1)_{em}} }
C
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S
{\displaystyle \mathbb {C\otimes S} }
idempotent projector
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+
=
1
/
2
(
1
+
i
e
15
)
{\displaystyle \rho _{+}=1/2(1+ie_{15})}
e
15
{\displaystyle e_{15}}
imaginary unit akin to
e
7
{\displaystyle e_{7}}
O
{\displaystyle \mathbb {O} }
Fano plane — that acts on the standard basis of the sedenions uniquely divides the algebra into three sets of split basis elements for
C
⊗
O
{\displaystyle \mathbb {C\otimes O} }
left actions on themselves generate three copies of the Clifford algebra
C
l
(
6
)
{\displaystyle \mathrm {C} l(6)}
minimal left ideals that describe a single generation of fermions with unbroken
S
U
(
3
)
c
×
U
(
1
)
e
m
{\displaystyle \mathrm {SU(3)_{c}\times U(1)_{em}} }
tensor products between normed division algebras generate zero divisors akin to those inside
S
{\displaystyle \mathbb {S} }
C
⊗
O
{\displaystyle \mathbb {C\otimes O} }
isomorphic to a Clifford algebra. Altogether, this permits three copies of
(
C
⊗
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)
L
≅
C
l
(
6
)
{\displaystyle (\mathbb {C\otimes O} )_{L}\cong \mathrm {Cl(6)} }
(
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⊗
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)
L
{\displaystyle \mathbb {(C\otimes S)} _{L}}
C
l
(
2
)
{\displaystyle \mathrm {C} l(2)}
CKM and PMNS matrices that, respectively, describe quark mixing and neutrino oscillations .
Sedenion neural networks provide [further explanation needed a means of efficient and compact expression in machine learning applications and have been used in solving multiple time-series and traffic forecasting problems.[3] [4]
^ Raoul E. Cawagas, et al. (2009). "THE BASIC SUBALGEBRA STRUCTURE OF THE CAYLEY-DICKSON ALGEBRA OF DIMENSION 32 (TRIGINTADUONIONS)" .
^ (Baez 2002 , p. 6)
^ Saoud, Lyes Saad; Al-Marzouqi, Hasan (2020). "Metacognitive Sedenion-Valued Neural Network and its Learning Algorithm" . IEEE Access . 8 : 144823–144838. doi :10.1109/ACCESS.2020.3014690 ISSN 2169-3536 . ^ Kopp, Michael; Kreil, David; Neun, Moritz; Jonietz, David; Martin, Henry; Herruzo, Pedro; Gruca, Aleksandra; Soleymani, Ali; Wu, Fanyou; Liu, Yang; Xu, Jingwei (2021-08-07). "Traffic4cast at NeurIPS 2020 – yet more on the unreasonable effectiveness of gridded geo-spatial processes" . NeurIPS 2020 Competition and Demonstration Track . PMLR: 325–343.
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 . New Series. 39 (2): 145–205. arXiv :math/0105155 doi :10.1090/S0273-0979-01-00934-X . MR 1886087 . S2CID 586512 .Biss, Daniel K.; Christensen, J. Daniel; Dugger, Daniel; Isaksen, Daniel C. (2007). "Large annihilators in Cayley-Dickson algebras II". Boletin de la Sociedad Matematica Mexicana . 3 : 269–292. arXiv :math/0702075 Bibcode :2007math......2075B . Guillard, Adam B.; Gresnigt, Niels G. (2019). "Three fermion generations with two unbroken gauge symmetries from the complex sedenions" . The European Physical Journal C 79 (5). Springer : 1–11 (446). arXiv :1904.03186 Bibcode :2019EPJC...79..446G . doi :10.1140/epjc/s10052-019-6967-1 . S2CID 102351250 . 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 CiteSeerX 10.1.1.240.6208 doi :10.1142/S0219498807001990 . S2CID 48162304 . 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", Bol. Soc. Mat. Mexicana , Series 3, 4 (1): 13–28, arXiv :q-alg/9710013 Bibcode :1997q.alg....10013G , 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 L. S. Saoud and H. Al-Marzouqi, "Metacognitive Sedenion-Valued Neural Network and its Learning Algorithm," in IEEE Access, vol. 8, pp. 144823-144838, 2020, doi:10.1109/ACCESS.2020.3014690 .
