"1,2,4,8 과 1,3,7"의 두 판 사이의 차이
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(피타고라스님이 이 페이지의 이름을 1248 과 137로 바꾸었습니다.) |
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* 복소수 | * 복소수 | ||
| − | * 외적 | + | * [[벡터의 외적(cross product)|외적]] |
* 사원수 | * 사원수 | ||
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<h5>참고할만한 자료</h5> | <h5>참고할만한 자료</h5> | ||
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* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
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<h5>관련논문</h5> | <h5>관련논문</h5> | ||
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| + | * [http://www.jstor.org/stable/2315620 The Scarcity of Cross Products on Euclidean Spaces]<br> | ||
| + | ** Bertram Walsh, <cite style="line-height: 2em;">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> | ||
| + | ** W. S. Massey, <cite style="line-height: 2em;">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> | ||
| + | ** J. F. Adams, <cite style="line-height: 2em;">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> | ||
| + | ** John Baez, 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> | ||
| + | ** Kenneth O. May, <cite style="line-height: 2em;">The American Mathematical Monthly</cite>, Vol. 73, No. 3 (Mar., 1966), pp. 289-291 | ||
2010년 8월 18일 (수) 09:45 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.
관련된 고교수학 또는 대학수학
- 복소수
- 외적
- 사원수
관련된 다른 주제들
- 해밀턴의 사원수
- Parallelizability of Spheres
- 호프 fibrations
관련도서 및 추천도서
- 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
- 도서검색
참고할만한 자료
- http://ko.wikipedia.org/wiki/
- 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/Composition_algebra
- http://en.wikipedia.org/wiki/Normed_division_algebra
- http://en.wikipedia.org/wiki/
- http://viswiki.com/en/
- http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
- http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
- 다음백과사전 http://enc.daum.net/dic100/search.do?q=
- 대한수학회 수학 학술 용어집
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=
블로그
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- 트렌비 블로그 검색 http://www.trenb.com/search.qst?q=
이미지 검색
- http://commons.wikimedia.org/w/index.php?title=Special%3ASearch&search=
- http://images.google.com/images?q=
- http://www.artchive.com