"1,2,4,8 과 1,3,7"의 두 판 사이의 차이

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10번째 줄: 10번째 줄:
 
 
 
 
  
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<h5>프로베니우스</h5>
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<h5>프로베니우스의 정리</h5>
  
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*  
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* 실수 위에 정의된 유한차원 associative division algebras
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* Frobenius’ theorem: any associative division algebra over R is isomorphic to R, C or H.
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Frobenius’ theorem: any associative division algebra over R is isomorphic to R, C or H.
 
  
 
 
 
 
  
 
<h5>Hurwitz's theorem for composition algebras (normed division algebras)</h5>
 
<h5>Hurwitz's theorem for composition algebras (normed division algebras)</h5>
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* 결합법칙을 가정하지 않는 경우
  
 
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:
109번째 줄: 110번째 줄:
 
** P. J. Hilton
 
** P. J. Hilton
 
* John H. Conway, Derek A. Smith On Quaternions and Octonions. A.K. Peters, 2003.
 
* John H. Conway, Derek A. Smith On Quaternions and Octonions. A.K. Peters, 2003.
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* John Baez, The Octonions, AMS 2001.
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* Division Algebras: Octonions, Quaternions, Complex Numbers, and the Algebraic Design of Physics<br>
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** Geoffrey Dixon
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** July 1994
 
*  도서내검색<br>
 
*  도서내검색<br>
 
** http://books.google.com/books?q=
 
** http://books.google.com/books?q=
130번째 줄: 133번째 줄:
 
** J. F. Adams
 
** J. F. Adams
 
** <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
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*  John Baez<br>
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** The Octonions, AMS 2001.
 
* http://ko.wikipedia.org/wiki/
 
* http://ko.wikipedia.org/wiki/
 
* http://en.wikipedia.org/wiki/Division_algebra
 
* http://en.wikipedia.org/wiki/Division_algebra

2009년 4월 17일 (금) 08:02 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. \(\iff\)\((p,q,r) = (0,1,1),(1,3,2),(3,7,4),(7,15,8)\)

 

 

프로베니우스의 정리
  • 실수 위에 정의된 유한차원 associative division algebras
  • Frobenius’ theorem: any associative division algebra over R is isomorphic to R, C or H.

 

Hurwitz's theorem for composition algebras (normed division algebras)
  • 결합법칙을 가정하지 않는 경우

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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