سدنيون

Sedenions
Sedenions
الرمز
Type	جبر غير تجميعي
الوحدات	e0...e15
المحايد الضربي	e0
الخصائص الرئيسية	تجميع القوى; توزيعية
الأنظمة الشائعة
	; ; ; ; ; ; الأنظمة الأقل شيوعاً أكتونيون (

في الجبر التجريدي، السـِدِنيون Sedenion يشكل 16 بعداً جبرياً فوق الأعداد الحقيقية. يرمز لمجموعة السدنيون بالرمز ${\displaystyle \mathbb {S$ . يعهد حالياً نوعان من السيدينيون:

سيدينيون تم الحصول عليه من إنشاء كايلي-ديكسون
سيدينيون مخروطي (ذو16 بعداً جبرياً).

الحساب

A visualization of a 4D extension to the cubic octonion, showing the 35 triads as hyperplanes through the real

{\displaystyle (e_{0 )

vertex of the sedenion example given. Note that the only exception is that the triple

{\displaystyle (e_{1 )

,

{\displaystyle (e_{2 )

,

{\displaystyle (e_{3 )

doesn't form a hyperplane with

{\displaystyle (e_{0 )

.

بشكل مشابه للأوكتونيون، فإن عملية ضرب السدنيون هي عملية غير تبديلية وغير تجميعية. ولكنه يمتلك خاصية تجميع القوى.

كل سدنيون هوتعبير عن هجريب خطي لعناصره وهي: 1, e₁, e₂, e₃, e₄, e₅, e₆, e₇, e₈, e₉, e₁₀, e₁₁, e₁₂, e₁₃, e₁₄ and e₁₅ والتي هي أسس الفضاء الشعاعي للسدنيون.

يعطى جدول ضرب عناصر السدنيون الستة عشرة على الشكل التالي:

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 ${\displaystyle e_{0$

{\displaystyle 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 (generated by ${\displaystyle e_{0$ to ${\displaystyle e_{7$ in the table below), and therefore also the quaternions (generated by ${\displaystyle e_{0$ to ${\displaystyle e_{3$ ), complex numbers (generated by ${\displaystyle e_{0$ and ${\displaystyle e_{1$ ) and reals (generated by ${\displaystyle e_{0$ ).

The sedenions have a multiplicative identity element ${\displaystyle e_{0$

جدول الضرب

		multiplier ${\displaystyle e_{j$
	${\displaystyle e_{i e_{j$	${\displaystyle e_{0$	${\displaystyle e_{1$	${\displaystyle e_{2$	${\displaystyle e_{3$	${\displaystyle e_{4$	${\displaystyle e_{5$	${\displaystyle e_{6$	${\displaystyle e_{7$	${\displaystyle e_{8$	${\displaystyle e_{9$	${\displaystyle e_{10$	${\displaystyle e_{11$	${\displaystyle e_{12$	${\displaystyle e_{13$	${\displaystyle e_{14$	${\displaystyle e_{15$
multiplicand ${\displaystyle e_{i$	${\displaystyle e_{0$	${\displaystyle e_{0$	${\displaystyle e_{1$	${\displaystyle e_{2$	${\displaystyle e_{3$	${\displaystyle e_{4$	${\displaystyle e_{5$	${\displaystyle e_{6$	${\displaystyle e_{7$	${\displaystyle e_{8$	${\displaystyle e_{9$	${\displaystyle e_{10$	${\displaystyle e_{11$	${\displaystyle e_{12$	${\displaystyle e_{13$	${\displaystyle e_{14$	${\displaystyle e_{15$
	${\displaystyle e_{1$	${\displaystyle e_{1$	− ${\displaystyle e_{0$	${\displaystyle e_{3$	− ${\displaystyle e_{2$	${\displaystyle e_{5$	− ${\displaystyle e_{4$	− ${\displaystyle e_{7$	${\displaystyle e_{6$	${\displaystyle e_{9$	− ${\displaystyle e_{8$	− ${\displaystyle e_{11$	${\displaystyle e_{10$	− ${\displaystyle e_{13$	${\displaystyle e_{12$	${\displaystyle e_{15$	− ${\displaystyle e_{14$
	${\displaystyle e_{2$	${\displaystyle e_{2$	− ${\displaystyle e_{3$	− ${\displaystyle e_{0$	${\displaystyle e_{1$	${\displaystyle e_{6$	${\displaystyle e_{7$	− ${\displaystyle e_{4$	− ${\displaystyle e_{5$	${\displaystyle e_{10$	${\displaystyle e_{11$	− ${\displaystyle e_{8$	− ${\displaystyle e_{9$	− ${\displaystyle e_{14$	− ${\displaystyle e_{15$	${\displaystyle e_{12$	${\displaystyle e_{13$
	${\displaystyle e_{3$	${\displaystyle e_{3$	${\displaystyle e_{2$	− ${\displaystyle e_{1$	− ${\displaystyle e_{0$	${\displaystyle e_{7$	− ${\displaystyle e_{6$	${\displaystyle e_{5$	− ${\displaystyle e_{4$	${\displaystyle e_{11$	− ${\displaystyle e_{10$	${\displaystyle e_{9$	− ${\displaystyle e_{8$	− ${\displaystyle e_{15$	${\displaystyle e_{14$	− ${\displaystyle e_{13$	${\displaystyle e_{12$
	${\displaystyle e_{4$	${\displaystyle e_{4$	− ${\displaystyle e_{5$	− ${\displaystyle e_{6$	− ${\displaystyle e_{7$	− ${\displaystyle e_{0$	${\displaystyle e_{1$	${\displaystyle e_{2$	${\displaystyle e_{3$	${\displaystyle e_{12$	${\displaystyle e_{13$	${\displaystyle e_{14$	${\displaystyle e_{15$	− ${\displaystyle e_{8$	− ${\displaystyle e_{9$	− ${\displaystyle e_{10$	− ${\displaystyle e_{11$
	${\displaystyle e_{5$	${\displaystyle e_{5$	${\displaystyle e_{4$	− ${\displaystyle e_{7$	${\displaystyle e_{6$	− ${\displaystyle e_{1$	− ${\displaystyle e_{0$	− ${\displaystyle e_{3$	${\displaystyle e_{2$	${\displaystyle e_{13$	− ${\displaystyle e_{12$	${\displaystyle e_{15$	− ${\displaystyle e_{14$	${\displaystyle e_{9$	− ${\displaystyle e_{8$	${\displaystyle e_{11$	− ${\displaystyle e_{10$
	${\displaystyle e_{6$	${\displaystyle e_{6$	${\displaystyle e_{7$	${\displaystyle e_{4$	− ${\displaystyle e_{5$	− ${\displaystyle e_{2$	${\displaystyle e_{3$	− ${\displaystyle e_{0$	− ${\displaystyle e_{1$	${\displaystyle e_{14$	− ${\displaystyle e_{15$	− ${\displaystyle e_{12$	${\displaystyle e_{13$	${\displaystyle e_{10$	− ${\displaystyle e_{11$	− ${\displaystyle e_{8$	${\displaystyle e_{9$
	${\displaystyle e_{7$	${\displaystyle e_{7$	− ${\displaystyle e_{6$	${\displaystyle e_{5$	${\displaystyle e_{4$	− ${\displaystyle e_{3$	− ${\displaystyle e_{2$	${\displaystyle e_{1$	− ${\displaystyle e_{0$	${\displaystyle e_{15$	${\displaystyle e_{14$	− ${\displaystyle e_{13$	− ${\displaystyle e_{12$	${\displaystyle e_{11$	${\displaystyle e_{10$	− ${\displaystyle e_{9$	− ${\displaystyle e_{8$
	${\displaystyle e_{8$	${\displaystyle e_{8$	− ${\displaystyle e_{9$	− ${\displaystyle e_{10$	− ${\displaystyle e_{11$	− ${\displaystyle e_{12$	− ${\displaystyle e_{13$	− ${\displaystyle e_{14$	− ${\displaystyle e_{15$	− ${\displaystyle e_{0$	${\displaystyle e_{1$	${\displaystyle e_{2$	${\displaystyle e_{3$	${\displaystyle e_{4$	${\displaystyle e_{5$	${\displaystyle e_{6$	${\displaystyle e_{7$
	${\displaystyle e_{9$	${\displaystyle e_{9$	${\displaystyle e_{8$	− ${\displaystyle e_{11$	${\displaystyle e_{10$	− ${\displaystyle e_{13$	${\displaystyle e_{12$	${\displaystyle e_{15$	− ${\displaystyle e_{14$	− ${\displaystyle e_{1$	− ${\displaystyle e_{0$	− ${\displaystyle e_{3$	${\displaystyle e_{2$	− ${\displaystyle e_{5$	${\displaystyle e_{4$	${\displaystyle e_{7$	− ${\displaystyle e_{6$
	${\displaystyle e_{10$	${\displaystyle e_{10$	${\displaystyle e_{11$	${\displaystyle e_{8$	− ${\displaystyle e_{9$	− ${\displaystyle e_{14$	− ${\displaystyle e_{15$	${\displaystyle e_{12$	${\displaystyle e_{13$	− ${\displaystyle e_{2$	${\displaystyle e_{3$	− ${\displaystyle e_{0$	− ${\displaystyle e_{1$	− ${\displaystyle e_{6$	− ${\displaystyle e_{7$	${\displaystyle e_{4$	${\displaystyle e_{5$
	${\displaystyle e_{11$	${\displaystyle e_{11$	− ${\displaystyle e_{10$	${\displaystyle e_{9$	${\displaystyle e_{8$	− ${\displaystyle e_{15$	${\displaystyle e_{14$	− ${\displaystyle e_{13$	${\displaystyle e_{12$	− ${\displaystyle e_{3$	− ${\displaystyle e_{2$	${\displaystyle e_{1$	− ${\displaystyle e_{0$	− ${\displaystyle e_{7$	${\displaystyle e_{6$	− ${\displaystyle e_{5$	${\displaystyle e_{4$
	${\displaystyle e_{12$	${\displaystyle e_{12$	${\displaystyle e_{13$	${\displaystyle e_{14$	${\displaystyle e_{15$	${\displaystyle e_{8$	− ${\displaystyle e_{9$	− ${\displaystyle e_{10$	− ${\displaystyle e_{11$	− ${\displaystyle e_{4$	${\displaystyle e_{5$	${\displaystyle e_{6$	${\displaystyle e_{7$	− ${\displaystyle e_{0$	− ${\displaystyle e_{1$	− ${\displaystyle e_{2$	− ${\displaystyle e_{3$
	${\displaystyle e_{13$	${\displaystyle e_{13$	− ${\displaystyle e_{12$	${\displaystyle e_{15$	− ${\displaystyle e_{14$	${\displaystyle e_{9$	${\displaystyle e_{8$	${\displaystyle e_{11$	− ${\displaystyle e_{10$	− ${\displaystyle e_{5$	− ${\displaystyle e_{4$	${\displaystyle e_{7$	− ${\displaystyle e_{6$	${\displaystyle e_{1$	− ${\displaystyle e_{0$	${\displaystyle e_{3$	− ${\displaystyle e_{2$
	${\displaystyle e_{14$	${\displaystyle e_{14$	− ${\displaystyle e_{15$	− ${\displaystyle e_{12$	${\displaystyle e_{13$	${\displaystyle e_{10$	− ${\displaystyle e_{11$	${\displaystyle e_{8$	${\displaystyle e_{9$	− ${\displaystyle e_{6$	− ${\displaystyle e_{7$	− ${\displaystyle e_{4$	${\displaystyle e_{5$	${\displaystyle e_{2$	− ${\displaystyle e_{3$	− ${\displaystyle e_{0$	${\displaystyle e_{1$
	${\displaystyle e_{15$	${\displaystyle e_{15$	${\displaystyle e_{14$	− ${\displaystyle e_{13$	− ${\displaystyle e_{12$	${\displaystyle e_{11$	${\displaystyle e_{10$	− ${\displaystyle e_{9$	${\displaystyle e_{8$	− ${\displaystyle e_{7$	${\displaystyle e_{6$	− ${\displaystyle e_{5$	− ${\displaystyle e_{4$	${\displaystyle e_{3$	${\displaystyle e_{2$	− ${\displaystyle e_{1$	− ${\displaystyle e_{0$

Sedenion properties

From the above table, we can see that:

{\displaystyle e_{0 e_{i =e_{i e_{0 =e_{i \,{\text{for all \,i,

{\displaystyle e_{i e_{i =-e_{0 \,\,{\text{for \,\,i\neq 0,

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

Anti-associative

The sedenions are not fully anti-associative. Choose any four generators, ${\displaystyle i,j,k$ and ${\displaystyle l$ . The following 5-cycle shows that at least one of these relations must associate.

${\displaystyle (ij)(kl)=-((ij)k)l=(i(jk))l=-i((jk)l)=i(j(kl))=-(ij)(kl)=0$

In particular, in the table above, using ${\displaystyle e_{1 ,e_{2 ,e_{4$ and ${\displaystyle e_{8$ the last expression associates. ${\displaystyle (e_{1 e_{2 )e_{12 =e_{1 (e_{2 e_{12 )=-e_{15$

Quaternionic subalgebras

The 35 triads that make up this specific sedenion multiplication table with theسبعة 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 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 ,
{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 { ${\displaystyle e_{a$ , ${\displaystyle e_{b$ , ${\displaystyle e_{c$ , ${\displaystyle e_{d$ , where ( ${\displaystyle e_{a$ + ${\displaystyle e_{b$ ) ${\displaystyle \circ$ ( ${\displaystyle e_{c$ + ${\displaystyle e_{d$ )=0:

التطبيقات

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 . (Note that in his paper, a "zero divisor" means a pair of elements that multiply to zero.)

انظر أيضاً

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

الهامش

^ (Baez 2002, p. 6)

المراجع

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
Baez, John C. (2002). "The Octonions". Bulletin of the American Mathematical Society. New Series. 39 (2): 145–205. arXiv:math/0105155. Bibcode:1994BAMaS..30..205W. 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. CiteSeerX 10.1.1.240.6208. 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", Bol. Soc. Mat. Mexicana, Series ثلاثة 4 (1): 13–28, Bibcode: 1997q.alg....10013G
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

نطقب:Number systems

[1] (Baez 2002, p. 6)

Sedenions
الرمز	${\displaystyle \mathbb {S$
Type	جبر غير تجميعي
الوحدات	e₀...e₁₅
المحايد الضربي	e₀
الخصائص الرئيسية	تجميع القوى توزيعية
الأنظمة الشائعة
${\displaystyle \mathbb {N$ ${\displaystyle \mathbb {Z$ ${\displaystyle \mathbb {Q$ ${\displaystyle \mathbb {R$ ${\displaystyle \mathbb {C$ ${\displaystyle \mathbb {H$ الأنظمة الأقل شيوعاً أكتونيون ( ${\displaystyle \mathbb {S$

سدنيون

سدنيون

الحساب

Sedenion properties

Anti-associative

Quaternionic subalgebras

التطبيقات

انظر أيضاً

الهامش

المراجع

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