Vizualizare sursă pentru „Sedenion”

În [[algebră abstractă |algebra abstractă]], '''sedenionii''' formează structuri algebrice cu 16 [[dimensiune (matematică) |dimensiuni]], și sunt obținuți prin aplicarea '''Construcției Cayley-Dickson''', studiate de Smith.<ref>Smith, 1995</ref> În general, sedenionii se notează cu <math>\mathbb{S}</math>.

== Sedenionii Cayley-Dickson ==
La fel ca [[octonion]]ii Cayley-Dickson, [[înmulțire (matematică) |înmulțirea]] sedenionilor Cayley-Dickson nu este nici [[comutativitate|comutativă]], nici [[asociativitate|asociativă]]. Dar, în comparație cu octonionii, sedenionii nu au proprietatea de a deveni alternativi. Totuși, ei au proprietatea unei puternice asociativități, care poate fi declarată pentru orice element <var>x</var> din <math>\mathbb{S}</math>, unde puterea <math>x^n</math> este [[expresie bine definită|bine definită]]. De asemenea, ei sunt și flexibili. Orice sedenion este o [[combinație liniară]] reală a [[unitate |unității]] 1, <var>e</var><sub>1</sub>, <var>e</var><sub>2</sub>, <var>e</var><sub>3</sub>, ..., și <var>e</var><sub>15</sub>,
care formează o [[bază (algebră liniară) |bază]] a [[spațiu vectorial |spațiului vectorial]] al sedenionilor.

Sedenionii au [[element neutru |elementul neutru]] multiplicativ 1 și nu au nici un divizor. Asta înseamnă că două numere nenule pot fi înmulțite pentru a obține zero: de exemplu (<var>e</var><sub>3</sub> + <var>e</var><sub>10</sub>)×(<var>e</var><sub>6</sub> − <var>e</var><sub>15</sub>). Toate sistemele de [[număr hipercomplex|numere hipercomplexe]] bazate pe construcția Cayley-Dickson de la sedenioni mai departe nu au nici un divizor.

[[Tabla înmulțirii]] a sedenionilor arată în felul următor:

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

== Aplicații ==
Moreno a arătat că spațiul cu norma 1 la sedenioni este homeomorf pentru forma compactă a Grupului Lie.<ref>Moreno, 1998</ref>

== Note ==
<references />

== Bibliografie ==
* {{en icon}} {{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}}
* {{en icon}} 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]
* {{en icon}} 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.
* {{en icon}} {{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}}
* {{en icon}} {{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}}

== Vezi și ==
* [[Număr hipercomplex]]
* [[Cuaternion]]
* [[Octonion]]
{{Portal|Matematică}}

[[Categorie:Numere]]