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

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

개요

루트 시스템은 유한차원 유클리드 벡터공간에서 여러가지 조건들을 만족시키는 벡터들의 모임이다
- non-zero eigenvalues of Cartan subalgebra
의 분류, 격자의 분류, 유한반사군과 콕세터군(finite reflection groups and Coxeter groups) 등에서 중요하게 활용
딘킨 다이어그램은 루트 시스템을 표현하는 그래프이다

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

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

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]

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

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

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

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

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

리 군

A_n SL_{n+1}(C)
B_n O_{2n+1}(C)
C_n Sp_{2n}(C)
D_n O_{2n}(C)

reflection groups

B_n, C_n, BC_n -> same reflection group (Z/nZ).S_n

역사

메모

\(\bullet - \bullet\)
http://demonstrations.wolfram.com/2DRootSystems/
reflection groups
lie algebras
Lie groups
algebraic groups
surfaces singularities
quiver
Platonic Solids

수학용어번역

사전 형태의 자료

http://ko.wikipedia.org/wiki/리대수