Hyperreal number

2014-01-26T03:46:07Z

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In [[abstract algebra]], '''sedenions''' form a 16-[[dimension of a vector space|dimensional]] non-associative [[algebra over a field|algebra]] over the [[real number|reals]] obtained by applying the [[Cayley–Dickson construction]] to the [[octonions]]. The set of '''sedenions''' is denoted by <math>\mathbb{S}</math>.

The term "sedenion" is also used for other 16-dimensional algebraic structures, such as a tensor product of 2 copies of the [[biquaternion]]s, or the algebra of 4 by 4 matrices over the reals, or that studied by {{harvtxt|Smith |1995}}.

== Cayley–Dickson Sedenions ==
===Arithmetic===
Like (Cayley–Dickson) [[octonion]]s, [[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 algebra|alternative]].
They do, however, have the property of [[power associativity]], which can be stated as for any element <var>x</var> of <math>\mathbb{S}</math>, the power <math>x^n</math> is well-defined. They are also [[Flexible identity|flexible]].

Every sedenion is a real [[linear combination]] of the unit sedenions 1, <var>e</var>1, <var>e</var>2, <var>e</var>3, ..., and <var>e</var>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 (<var>e</var>3 + <var>e</var>10)×(<var>e</var>6 − <var>e</var>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:

{| class="wikitable" style="text-align: center;"
|-
! ×
! 1
! <var>e</var>1
! <var>e</var>2
! <var>e</var>3
! <var>e</var>4
! <var>e</var>5
! <var>e</var>6
! <var>e</var>7
! <var>e</var>8
! <var>e</var>9
! <var>e</var>10
! <var>e</var>11
! <var>e</var>12
! <var>e</var>13
! <var>e</var>14
! <var>e</var>15
|-
! 1
| 1
| <var>e</var>1
| <var>e</var>2
| <var>e</var>3
| <var>e</var>4
| <var>e</var>5
| <var>e</var>6
| <var>e</var>7
| <var>e</var>8
| <var>e</var>9
| <var>e</var>10
| <var>e</var>11
| <var>e</var>12
| <var>e</var>13
| <var>e</var>14
| <var>e</var>15
|-
! <var>e</var>1
| <var>e</var>1
| −1
| <var>e</var>3
| −<var>e</var>2
| <var>e</var>5
| −<var>e</var>4
| −<var>e</var>7
| <var>e</var>6
| <var>e</var>9
| −<var>e</var>8
| −<var>e</var>11
| <var>e</var>10
| −<var>e</var>13
| <var>e</var>12
| <var>e</var>15
| −<var>e</var>14
|-
! <var>e</var>2
| <var>e</var>2
| −<var>e</var>3
| −1
| <var>e</var>1
| <var>e</var>6
| <var>e</var>7
| −<var>e</var>4
| −<var>e</var>5
| <var>e</var>10
| <var>e</var>11
| −<var>e</var>8
| −<var>e</var>9
| −<var>e</var>14
| −<var>e</var>15
| <var>e</var>12
| <var>e</var>13
|-
! <var>e</var>3
| <var>e</var>3
| <var>e</var>2
| −<var>e</var>1
| −1
| <var>e</var>7
| −<var>e</var>6
| <var>e</var>5
| −<var>e</var>4
| <var>e</var>11
| −<var>e</var>10
| <var>e</var>9
| −<var>e</var>8
| −<var>e</var>15
| <var>e</var>14
| −<var>e</var>13
| <var>e</var>12
|-
! <var>e</var>4
| <var>e</var>4
| −<var>e</var>5
| −<var>e</var>6
| −<var>e</var>7
| −1
| <var>e</var>1
| <var>e</var>2
| <var>e</var>3
| <var>e</var>12
| <var>e</var>13
| <var>e</var>14
| <var>e</var>15
| −<var>e</var>8
| −<var>e</var>9
| −<var>e</var>10
| −<var>e</var>11
|-
! <var>e</var>5
| <var>e</var>5
| <var>e</var>4
| −<var>e</var>7
| <var>e</var>6
| −<var>e</var>1
| −1
| −<var>e</var>3
| <var>e</var>2
| <var>e</var>13
| −<var>e</var>12
| <var>e</var>15
| −<var>e</var>14
| <var>e</var>9
| −<var>e</var>8
| <var>e</var>11
| −<var>e</var>10
|-
! <var>e</var>6
| <var>e</var>6
| <var>e</var>7
| <var>e</var>4
| −<var>e</var>5
| −<var>e</var>2
| <var>e</var>3
| −1
| −<var>e</var>1
| <var>e</var>14
| −<var>e</var>15
| −<var>e</var>12
| <var>e</var>13
| <var>e</var>10
| −<var>e</var>11
| −<var>e</var>8
| <var>e</var>9
|-
! <var>e</var>7
| <var>e</var>7
| −<var>e</var>6
| <var>e</var>5
| <var>e</var>4
| −<var>e</var>3
| −<var>e</var>2
| <var>e</var>1
| −1
| <var>e</var>15
| <var>e</var>14
| −<var>e</var>13
| −<var>e</var>12
| <var>e</var>11
| <var>e</var>10
| −<var>e</var>9
| −<var>e</var>8
|-
! <var>e</var>8
| <var>e</var>8
| −<var>e</var>9
| −<var>e</var>10
| −<var>e</var>11
| −<var>e</var>12
| −<var>e</var>13
| −<var>e</var>14
| −<var>e</var>15
| −1
| <var>e</var>1
| <var>e</var>2
| <var>e</var>3
| <var>e</var>4
| <var>e</var>5
| <var>e</var>6
| <var>e</var>7
|-
! <var>e</var>9
| <var>e</var>9
| <var>e</var>8
| −<var>e</var>11
| <var>e</var>10
| −<var>e</var>13
| <var>e</var>12
| <var>e</var>15
| −<var>e</var>14
| −<var>e</var>1
| −1
| −<var>e</var>3
| <var>e</var>2
| −<var>e</var>5
| <var>e</var>4
| <var>e</var>7
| −<var>e</var>6
|-
! <var>e</var>10
| <var>e</var>10
| <var>e</var>11
| <var>e</var>8
| −<var>e</var>9
| −<var>e</var>14
| −<var>e</var>15
| <var>e</var>12
| <var>e</var>13
| −<var>e</var>2
| <var>e</var>3
| −1
| −<var>e</var>1
| −<var>e</var>6
| −<var>e</var>7
| <var>e</var>4
| <var>e</var>5
|-
! <var>e</var>11
| <var>e</var>11
| −<var>e</var>10
| <var>e</var>9
| <var>e</var>8
| −<var>e</var>15
| <var>e</var>14
| −<var>e</var>13
| <var>e</var>12
| −<var>e</var>3
| −<var>e</var>2
| <var>e</var>1
| −1
| −<var>e</var>7
| <var>e</var>6
| −<var>e</var>5
| <var>e</var>4
|-
! <var>e</var>12
| <var>e</var>12
| <var>e</var>13
| <var>e</var>14
| <var>e</var>15
| <var>e</var>8
| −<var>e</var>9
| −<var>e</var>10
| −<var>e</var>11
| −<var>e</var>4
| <var>e</var>5
| <var>e</var>6
| <var>e</var>7
| −1
| −<var>e</var>1
| −<var>e</var>2
| −<var>e</var>3
|-
! <var>e</var>13
| <var>e</var>13
| −<var>e</var>12
| <var>e</var>15
| −<var>e</var>14
| <var>e</var>9
| <var>e</var>8
| <var>e</var>11
| −<var>e</var>10
| −<var>e</var>5
| −<var>e</var>4
| <var>e</var>7
| −<var>e</var>6
| <var>e</var>1
| −1
| <var>e</var>3
| −<var>e</var>2
|-
! <var>e</var>14
| <var>e</var>14
| −<var>e</var>15
| −<var>e</var>12
| <var>e</var>13
| <var>e</var>10
| −<var>e</var>11
| <var>e</var>8
| <var>e</var>9
| −<var>e</var>6
| −<var>e</var>7
| −<var>e</var>4
| <var>e</var>5
| <var>e</var>2
| −<var>e</var>3
| −1
| <var>e</var>1
|-
! <var>e</var>15
| <var>e</var>15
| <var>e</var>14
| −<var>e</var>13
| −<var>e</var>12
| <var>e</var>11
| <var>e</var>10
| −<var>e</var>9
| <var>e</var>8
| −<var>e</var>7
| <var>e</var>6
| −<var>e</var>5
| −<var>e</var>4
| <var>e</var>3
| <var>e</var>2
| −<var>e</var>1
| −1
|}

==Applications==
{{harvtxt|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 (mathematics)|G2]].

==See also==
* [[Pfister's sixteen-square identity]]
* [[Hypercomplex number]]
* [[Split-complex number]]

==References==

*{{Citation | last1=Imaeda | first1=K. | last2=Imaeda | first2=M. | title=Sedenions: algebra and analysis | doi=10.1016/S0096-3003(99)00140-X | mr=1786945 | year=2000 | journal=Applied mathematics and computation | volume=115 | issue=2 | pages=77–88}}
* 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]
* Kivunge, Benard M. and Smith, Jonathan D. H: "[http://www.emis.de/journals/CMUC/pdf/cmuc0402/kivunge.pdf Subloops of sedenions]", Comment.Math.Univ.Carolinae 45,2 (2004)295–302.
*{{Citation | last1=Moreno | first1=Guillermo | title=The zero divisors of the Cayley–Dickson algebras over the real numbers | arxiv=q-alg/9710013 | mr=1625585 | year=1998 | journal=Sociedad Matemática Mexicana. Boletí n. Tercera Serie | volume=4 | issue=1 | pages=13–28}}
*{{Citation | last1=Smith | first1=Jonathan D. H. | title=A left loop on the 15-sphere | doi=10.1006/jabr.1995.1237 | mr=1345298 | year=1995 | journal=[[Journal of Algebra]] | volume=176 | issue=1 | pages=128–138}}
{{Number Systems}}

[[Category:Hypercomplex numbers]]
[[Category:Non-associative algebras]]

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