"1,2,4,8 과 1,3,7"의 두 판 사이의 차이

수학노트
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21번째 줄: 21번째 줄:
 
* 결합법칙을 가정하지 않는 경우
 
* 결합법칙을 가정하지 않는 경우
  
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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:
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a '''normed division algebra'''<em style="">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:
  
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: <math>\|xy\| = \|x\| \|y\|</math> for all <em>x</em> and <em>y</em> in <em>A</em>.
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: <math>\|xy\| = \|x\| \|y\|</math> for all <em style="">x</em> and <em style="">y</em> in <em style="">A</em>.
  
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<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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<br>'''composition algebra'''<em style="">A</em> over a [http://en.wikipedia.org/wiki/Field_%28mathematics%29 field]<em style="">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 style="">K</em> together with a [http://en.wikipedia.org/wiki/Nondegenerate nondegenerate][http://en.wikipedia.org/wiki/Quadratic_form quadratic form]<em style="">N</em> which satisfies
  
 
: <math>N(xy) = N(x)N(y)\,</math>
 
: <math>N(xy) = N(x)N(y)\,</math>
  
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for all <em>x</em> and <em>y</em> in <em>A</em>.
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for all <em style="">x</em> and <em style="">y</em> in <em style="">A</em>.
  
 
 
 
 
110번째 줄: 110번째 줄:
 
** July 1994
 
** July 1994
 
* [http://www.math.cornell.edu/%7Ehatcher/VBKT/VBpage.html Vector Bundles & K-Theory]<br>
 
* [http://www.math.cornell.edu/%7Ehatcher/VBKT/VBpage.html Vector Bundles & K-Theory]<br>
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**  
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** Allen Hatcher
 
*  도서내검색<br>
 
*  도서내검색<br>
 
** http://books.google.com/books?q=
 
** http://books.google.com/books?q=
123번째 줄: 123번째 줄:
  
 
* [http://www.jstor.org/stable/2315620 The Scarcity of Cross Products on Euclidean Spaces]<br>
 
* [http://www.jstor.org/stable/2315620 The Scarcity of Cross Products on Euclidean Spaces]<br>
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** Bertram Walsh
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** Bertram Walsh, <cite>The American Mathematical Monthly</cite>, Vol. 74, No. 2 (Feb., 1967), pp. 188-194
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** <cite>The American Mathematical Monthly</cite>, Vol. 74, No. 2 (Feb., 1967), pp. 188-194
 
 
* [http://www.jstor.org/stable/2323537 Cross Products of Vectors in Higher Dimensional Euclidean Spaces]<br>
 
* [http://www.jstor.org/stable/2323537 Cross Products of Vectors in Higher Dimensional Euclidean Spaces]<br>
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** W. S. Massey
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** W. S. Massey, <cite>The American Mathematical Monthly</cite>, Vol. 90, No. 10 (Dec., 1983), pp. 697-701
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** <cite>The American Mathematical Monthly</cite>, Vol. 90, No. 10 (Dec., 1983), pp. 697-701
 
 
* [http://www.jstor.org/stable/1970147 On the Non-Existence of Elements of Hopf Invariant One]<br>
 
* [http://www.jstor.org/stable/1970147 On the Non-Existence of Elements of Hopf Invariant One]<br>
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** J. F. Adams
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** J. F. Adams, <cite>The Annals of Mathematics</cite>, Second Series, Vol. 72, No. 1 (Jul., 1960), pp. 20-104
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** <cite>The Annals of Mathematics</cite>, Second Series, Vol. 72, No. 1 (Jul., 1960), pp. 20-104
 
 
* [http://math.ucr.edu/home/baez/octonions/ The Octonions]<br>
 
* [http://math.ucr.edu/home/baez/octonions/ The Octonions]<br>
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** John Baez
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** John Baez, AMS 2001
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** AMS 2001
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* [http://www.jstor.org/stable/2315349 The Impossibility of a Division Algebra of Vectors in Three Dimensional Space]<br>
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** Kenneth O. May, <cite style="line-height: 2em;">The American Mathematical Monthly</cite>, Vol. 73, No. 3 (Mar., 1966), pp. 289-291
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* http://ko.wikipedia.org/wiki/
 
* http://ko.wikipedia.org/wiki/
 
* http://en.wikipedia.org/wiki/Division_algebra
 
* http://en.wikipedia.org/wiki/Division_algebra
145번째 줄: 145번째 줄:
 
* 다음백과사전 http://enc.daum.net/dic100/search.do?q=
 
* 다음백과사전 http://enc.daum.net/dic100/search.do?q=
 
* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]
 
* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]
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<h5>관련논문</h5>
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2009년 8월 25일 (화) 15:26 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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