루트 시스템 (root system)과 딘킨 다이어그램 (Dynkin diagram)
이 항목의 스프링노트 원문주소==
개요==
- 루트 시스템은 유한차원 유클리드 벡터공간에서 여러가지 조건들을 만족시키는 벡터들의 모임이다
- non-zero eigenvalues of Cartan subalgebra
- 리군과 리대수의 분류, 격자의 분류, 유한반사군과 콕세터군(finite reflection groups and Coxeter groups) 등에서 중요하게 활용
- 딘킨 다이어그램은 루트 시스템을 표현하는 그래프이다
- non-zero eigenvalues of Cartan subalgebra
정의==
- E를 내적이 주어진 유클리드 벡터공간이라 하자.
- 다음 조건을 만족시키는 E의 유한인 부분집합 \(\Phi\)를 루트 시스템이라 한다.
- \(\Phi\)는 E를 스팬(span)하며 \(0 \not \in \Phi\)
- (reduced) \(\alpha \in \Phi\), \(\lambda \alpha \in \Phi \iff \lambda=\pm 1\)
- \(\alpha,\beta \in \Phi\)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 crystallographic 또는 integraliy 조건이라 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
- \(\Phi\)는 E를 스팬(span)하며 \(0 \not \in \Phi\)
- (reduced) \(\alpha \in \Phi\), \(\lambda \alpha \in \Phi \iff \lambda=\pm 1\)
- \(\alpha,\beta \in \Phi\)이면 \(\sigma_\alpha(\beta) =\beta-2\frac{(\beta,\alpha)}{(\alpha,\alpha)}\alpha \in \Phi\)
- \(\langle \beta, \alpha \rangle = 2 \frac{(\beta,\alpha)}{(\alpha,\alpha)} \in \mathbb{Z}\)
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
\(\frac{\pi}{2}\) , \(\frac{\pi}{3}\), \(\frac{\pi}{4}\), \(\frac{\pi}{6}\)
0,1,2,3 lines
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
[1]
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[/pages/2696052/attachments/2088321 MSP402197733f5dbe80g5d000056hb767e4digb412.gif]
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ADE 의 분류==
(0) G cannot contain affine A_n, D_n, E_n
(1) G is a tree (contains no cycles = affine A_n)
(2) G has \leq 1 branch point (does not contain affine D_5, D_6,D_7, )
(3) branch point has order \leq 3 (affine D_4)
What are length of legs of G?
Leg of length 0 -> G=A_n
so assume legs have length \geq 1
(4) Not all legs have length \geq 2 : cannot contain affine E_6
so one leg has length 1
2 legs of length 1 : G is D_n
so can assume 2 other legs have length \geq 2
(5) cannot have 2 legs length \geq 3 because of affine E_7
So G has 1 leg length 1, 1 of length 2, one of length \geq 2
length is \leq 4, as G does not contain affine E_8
So G is E6,E7, E8
일반적인 경우
- 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
- 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