1,2,4,8 과 1,3,7

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 결합법칙을 만족하는 유한차원 division algebras
• 프로베니우스의 정리
  any associative division algebra over R is isomorphic to R, C or H.
  

 

composition 대수에 관한 후르비츠의 정리 (normed division algebras)

• 결합법칙을 가정하지 않는 경우
• 실수나 복소수위에 정의된

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.

 

 

관련된 고교수학 또는 대학수학

• 복소수
• 외적
• 사원수

 

 

관련된 항목들

• 해밀턴의 사원수
• Parallelizability of Spheres
• 호프 fibrations

 

 

수학용어번역

• 단어사전 http://www.google.com/dictionary?langpair=en%7Cko&q=
• 발음사전 http://www.forvo.com/search/
• 대한수학회 수학 학술 용어집
  
  • http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=norm
• 남·북한수학용어비교
• 대한수학회 수학용어한글화 게시판

 

 

 

 

 

관련도서

• General Cohomology Theory and K-Theory (London Mathematical Society Lecture Note Series) (Paperback)
  
  • P. J. Hilton
• On Quaternions and Octonions
  
  • John H. Conway, Derek A. Smith, A.K. Peters, 2003.
• Division Algebras: Octonions, Quaternions, Complex Numbers, and the Algebraic Design of Physics
  
  • Geoffrey Dixon, July 1994
• Vector Bundles & K-Theory
  
  • Allen Hatcher
• 도서내검색
  
  • http://books.google.com/books?q=
  • http://book.daum.net/search/contentSearch.do?query=
• 도서검색
  
  • http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
  • http://book.daum.net/search/mainSearch.do?query=

 

사전형태의 자료

• http://ko.wikipedia.org/wiki/
• [1]http://en.wikipedia.org/wiki/Division_algebra
• http://en.wikipedia.org/wiki/Hurwitz's_theorem_(normed_division_algebras)
• http://en.wikipedia.org/wiki/Composition_algebra
• http://en.wikipedia.org/wiki/Normed_division_algebra

 

 

관련논문

• The Scarcity of Cross Products on Euclidean Spaces
  
  • Bertram Walsh, The American Mathematical Monthly, Vol. 74, No. 2 (Feb., 1967), pp. 188-194
• Cross Products of Vectors in Higher Dimensional Euclidean Spaces
  
  • W. S. Massey, The American Mathematical Monthly, Vol. 90, No. 10 (Dec., 1983), pp. 697-701
• On the Non-Existence of Elements of Hopf Invariant One
  
  • J. F. Adams, The Annals of Mathematics, Second Series, Vol. 72, No. 1 (Jul., 1960), pp. 20-104
• The Octonions
  
  • John Baez, AMS 2001

 

• The Impossibility of a Division Algebra of Vectors in Three Dimensional Space
  
  • Kenneth O. May, The American Mathematical Monthly, Vol. 73, No. 3 (Mar., 1966), pp. 289-291

 

 

관련기사

• 네이버 뉴스 검색 (키워드 수정)
  
  • http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
  • http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
  • http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
  • http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
  • http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=