"1,2,4,8 과 1,3,7"의 두 판 사이의 차이

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<h5>Hurwitz's theorem for composition algebras (Normed division algebras)</h5>
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<h5>프로베니우스</h5>
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*  
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Frobenius’ theorem: any associative division algebra over R is isomorphic to R, C or H.
  
 
 
 
 
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<h5>Hurwitz's theorem for composition algebras (normed division algebras)</h5>
  
 
a '''normed division algebra'''<em>A</em> is a [http://en.wikipedia.org/wiki/Division_algebra division algebra] over the [http://en.wikipedia.org/wiki/Real_number real] or [http://en.wikipedia.org/wiki/Complex_number complex] numbers which is also a [http://en.wikipedia.org/wiki/Normed_vector_space normed vector space], with norm || · || satisfying the following property:
 
a '''normed division algebra'''<em>A</em> is a [http://en.wikipedia.org/wiki/Division_algebra division algebra] over the [http://en.wikipedia.org/wiki/Real_number real] or [http://en.wikipedia.org/wiki/Complex_number complex] numbers which is also a [http://en.wikipedia.org/wiki/Normed_vector_space normed vector space], with norm || · || satisfying the following property:
  
 
: <math>\|xy\| = \|x\| \|y\|</math> for all <em>x</em> and <em>y</em> in <em>A</em>.
 
: <math>\|xy\| = \|x\| \|y\|</math> for all <em>x</em> and <em>y</em> in <em>A</em>.
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Normed division algebras are a special case of composition algebras
 
  
 
<br>'''composition algebra'''<em>A</em> over a [http://en.wikipedia.org/wiki/Field_%28mathematics%29 field]<em>K</em> is a [http://en.wikipedia.org/wiki/Unital unital] (but not necessarily [http://en.wikipedia.org/wiki/Associative associative]) [http://en.wikipedia.org/wiki/Algebra_over_a_field algebra] over <em>K</em> together with a [http://en.wikipedia.org/wiki/Nondegenerate nondegenerate][http://en.wikipedia.org/wiki/Quadratic_form quadratic form]<em>N</em> which satisfies
 
<br>'''composition algebra'''<em>A</em> over a [http://en.wikipedia.org/wiki/Field_%28mathematics%29 field]<em>K</em> is a [http://en.wikipedia.org/wiki/Unital unital] (but not necessarily [http://en.wikipedia.org/wiki/Associative associative]) [http://en.wikipedia.org/wiki/Algebra_over_a_field algebra] over <em>K</em> together with a [http://en.wikipedia.org/wiki/Nondegenerate nondegenerate][http://en.wikipedia.org/wiki/Quadratic_form quadratic form]<em>N</em> which satisfies
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Normed division algebras are a special case of composition algebras
  
 
 
 
 
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(정리) Hurwitz
 
(정리) Hurwitz
  
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the only composition algebras over <math>\Bbb{R}</math> are <math>\Bbb{R}</math>, <math>\Bbb{R}</math>, <math>\Bbb{R}</math>, <math>\Bbb{R}</math>\Bbb{R} , \mathbb{C}, \mathbb H and \mathbb{O}
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The only composition algebras over <math>\Bbb{R}</math> are <math>\Bbb{R}</math>,<math>\Bbb{C}</math>, <math>\Bbb{H}</math>, and <math>\Bbb{O}</math> , that is the real numbers, the complex numbers, the quaternions and the octonions.
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that is the real numbers, the complex numbers, the quaternions and the octonions.
 
  
 
 
 
 
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* [[해밀턴의 사원수(quarternions)|해밀턴의 사원수]]
 
* [[해밀턴의 사원수(quarternions)|해밀턴의 사원수]]
 
* Parallelizability of Spheres
 
* Parallelizability of Spheres
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* 호프 fibrations
  
 
 
 
 
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** <cite>The Annals of Mathematics</cite>, Second Series, Vol. 72, No. 1 (Jul., 1960), pp. 20-104
 
** <cite>The Annals of Mathematics</cite>, Second Series, Vol. 72, No. 1 (Jul., 1960), pp. 20-104
 
* http://ko.wikipedia.org/wiki/
 
* http://ko.wikipedia.org/wiki/
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* http://en.wikipedia.org/wiki/Division_algebra
 
* http://en.wikipedia.org/wiki/Hurwitz%27s_theorem#Hurwitz.27s_theorem_for_composition_algebras
 
* http://en.wikipedia.org/wiki/Hurwitz%27s_theorem#Hurwitz.27s_theorem_for_composition_algebras
 
* http://en.wikipedia.org/wiki/Composition_algebra
 
* http://en.wikipedia.org/wiki/Composition_algebra

2009년 4월 17일 (금) 07:55 판

간단한 소개
  • \(\mathbb R^n\) 은 division algebra이다 \(\iff\)\(n=1,2,4,8\)
  • \(S^n\) 는 H-space 이다. \(\iff\)\(n=0,1,3,7\)
  • \(S^n\) 은 n개의 일차독립인 벡터장을 갖는다 \(\iff\)\(n=0,1,3,7\)
  • fiber 번들 \(S^p \to S^q \to S^r\) 이 존재한다. \(\iff\)\((p,q,r) = (0,1,1),(1,3,2),(3,7,4),(7,15,8)\)

 

 

프로베니우스
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Frobenius’ theorem: any associative division algebra over R is isomorphic to R, C or H.

 

Hurwitz's theorem for composition algebras (normed division algebras)

a normed division algebraA is a division algebra over the real or complex numbers which is also a normed vector space, with norm || · || satisfying the following property:

\[\|xy\| = \|x\| \|y\|\] for all x and y in A.


composition algebraA over a fieldK is a unital (but not necessarily associative) algebra over K together with a nondegeneratequadratic formN which satisfies

\[N(xy) = N(x)N(y)\,\]

for all x and y in A.

 

Normed division algebras are a special case of composition algebras

 

(정리) Hurwitz

The only composition algebras over \(\Bbb{R}\) are \(\Bbb{R}\),\(\Bbb{C}\), \(\Bbb{H}\), and \(\Bbb{O}\) , that is the real numbers, the complex numbers, the quaternions and the octonions.

 

 

 

 


 

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