루트 시스템 (root system)과 딘킨 다이어그램 (Dynkin diagram)

 

 

개요

• 루트 시스템은 유한차원 유클리드 벡터공간에서 여러가지 조건들을 만족시키는 벡터들의 모임이다
   
  
• 리군과 리대수의 분류, 격자의 분류, 유한반사군과 콕세터군(finite reflection groups and Coxeter groups) 등에서 중요하게 활용
  
• 딘킨 다이어그램의 분류
  

 

 

정의

• E를 내적이 주어진 유클리드 벡터공간이라 하자.
• 다음 조건을 만족시키는 E의 유한인 부분집합 \(\Phi\)를 루트 시스템이라 한다.
  
  •  \(\Phi\)는 E를 스팬(span)하며 \(0 \not \in \Phi\)
  • \(\alpha \in \Phi\), \(\lambda \alpha \in \Phi \iff \lambda=\pm 1\)
  • \(\alpha,\beta \in \Phi\)이면   \(\sigma_\alpha(\beta) =\beta-2\frac{(\beta,\alpha)}{(\alpha,\alpha)}\alpha \in \Phi\)
  • \(\langle \beta, \alpha \rangle = 2 \frac{(\beta,\alpha)}{(\alpha,\alpha)} \in \mathbb{Z}\)
• 마지막 조건을 crystallographic조건이라 한다
• a subgroup of \(GL(V)\) is crystallographic if it stabilizes a lattice L in V
  
• e.g. the Weyl group of a Lie algebra stabilizes the root lattice or the weight lattice

 

 

딘킨 다이어그램 (Dynkin diagram)

• first draw the simple roots as nodes
• draw \(4(e_i, e_j)^2\)lines for two roots \(e_i, e_j\)
  \(\frac{\pi}{2}\) , \(\frac{\pi}{3}\), \(\frac{\pi}{4}\), \(\frac{\pi}{6}\)
  0,1,2,3 lines
  
• how to classify all connected admissible diagrams
  
  • subdiagram is also admissible
  • there are at most (n-1) pairs of nodes
  • no node has more than 3 lines
  • study double lines and triple nodes

 

 

 

2차원 루트 시스템의 분류

• \(A_1\times A_1\), \(A_2\), \(B_2\), \(G_2\)
  

A1 x A1

http://www.wolframalpha.com/input/?i=r%3D1%2Bcos+(4theta)

A2

http://www.wolframalpha.com/input/?i=r%3D1%2B+cos+(6theta)

B2

http://www.wolframalpha.com/input/?i=r%3D1-+(sqrt2+%2B1)^2+cos+(4theta)

G2

http://www.wolframalpha.com/input/?i=r%3D1-(sqrt+3+%2B1)^2cos+(6theta)/2

 

http://en.wikipedia.org/wiki/Root_system

[/pages/2696052/attachments/2088323 MSP45719773453e5409bcd000043c1iebh17cda58g.gif]

[/pages/2696052/attachments/2088321 MSP402197733f5dbe80g5d000056hb767e4digb412.gif]

[/pages/2696052/attachments/2088319 MSP132719772cfcfe659i75000064ieda8fh9d30h5e.gif]

[/pages/2696052/attachments/2088317 MSP98119772g2ig5gid8he000031i1h30a8gacdi00.gif]

 

 

재미있는 사실

 

• Math Overflow http://mathoverflow.net/search?q=
• 네이버 지식인 http://kin.search.naver.com/search.naver?where=kin_qna&query=

 

 

역사

 

• http://www.google.com/search?hl=en&tbs=tl:1&q=
• 수학사연표
•  

 

 

메모

• reflection groups
• lie algebras
• Lie groups
• algebraic groups
• surfaces singularities
• quiver
• Platonic Solids

 

 

 

 

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=
• 남·북한수학용어비교
• 대한수학회 수학용어한글화 게시판

 

 

사전 형태의 자료

• http://ko.wikipedia.org/wiki/리대수

• http://en.wikipedia.org/wiki/root_systems
• http://en.wikipedia.org/wiki/Dynkin_diagram
• http://en.wikipedia.org/wiki/Coxeter_number

• http://en.wikipedia.org/wiki/
• http://www.wolframalpha.com/input/?i=
• NIST Digital Library of Mathematical Functions
• The On-Line Encyclopedia of Integer Sequences
  
  • http://www.research.att.com/~njas/sequences/?q=

 

 

관련논문

• Two Amusing Dynkin Diagram Graph Classifications
  
  • Robert A. Proctor, The American Mathematical Monthly, Vol. 100, No. 10 (Dec., 1993), pp. 937-941
• http://www.jstor.org/action/doBasicSearch?Query=
• http://www.ams.org/mathscinet
• http://dx.doi.org/

 

 

관련도서

• 도서내검색
  
  • http://books.google.com/books?q=
  • http://book.daum.net/search/contentSearch.do?query=
• 도서검색
  
  • http://books.google.com/books?q=
  • http://book.daum.net/search/mainSearch.do?query=
  • http://book.daum.net/search/mainSearch.do?query=

 

 

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=

 

 

블로그

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
• 네이버 오늘의과학
• 수학동아
• Mathematical Moments from the AMS
• BetterExplained