"유한바일군의 계산 강의노트"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) |
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| 12번째 줄: | 12번째 줄: | ||
* how to generate all positive roots | * how to generate all positive roots | ||
* how to generate elements of the Weyl group | * how to generate elements of the Weyl group | ||
| + | |||
| + | |||
| + | ==background== | ||
| + | ===simple Lie algebras=== | ||
| + | * Lie algebra : vector space with a bilinear, alternating product | ||
| + | $$ | ||
| + | [\,,\,]: \mathfrak{g}\times \mathfrak{g} \to \mathfrak{g} | ||
| + | $$ | ||
| + | satisfying the Jacobi identity | ||
| + | :<math>[a, [b,c]]+[b,[c,a]]+[c,[a,b]]=0</math> | ||
| + | * $\mathfrak{sl}_2$ : $2\times 2$ matrix with trace 0 over $\mathbb{C}$ with commutator $[a,b]=ab-ba$ | ||
| + | * basis <math>\langle e,f,h \rangle</math> | ||
| + | :<math>e=\begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}, f=\begin{pmatrix} 0&0\\ 1&0 \end{pmatrix}, h=\begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}</math> | ||
| + | :<math>[h,e]=2e, [h,f]=-2f,[e,f]=h</math> | ||
| + | ===Cartan-Killing=== | ||
| + | * classification of finite-dim'l simple Lie algebras over $\mathbb{C}$ | ||
| + | * key idea : use linear algebra via adjoint representation | ||
| + | * decomposition of $\mathfrak{g}$ relative to a maximal abelian subalgebra $\mathfrak{h}$ -> root space decomposition | ||
| + | * key structure : root systems (highly constrained combinatorial object), $A_2$ example | ||
| + | * possible root system of a simple Lie algebra : $A_l,B_l,C_l,D_l,E_6,E_7,E_8,F_4,G_2$ | ||
| + | * this can be compactly encoded in Cartan matrix or Dynkin diagram | ||
| + | |||
| + | ===Cartan-Weyl=== | ||
| + | * classification of finite-dim'l irr. rep'n | ||
| + | * key concept : weight space decomposition of rep'n | ||
| + | * Cartan : dominant integral highest weight - finite-dim'l irr. rep'n (weights in the fundamental chamber, $A_2$) | ||
| + | * character of a representaion : generating function of dimension of each weight space | ||
| + | $$\operatorname{ch}(V):=\sum_{\mu \in \mathfrak{h}^{*}} (\dim{V_{\mu}})e^{\mu}$$ | ||
| + | * Weyl : character formula, of irr. rep'n <math>V=L(\lambda)</math> with highest weight $\lambda$ | ||
| + | $$ | ||
| + | \begin{align} | ||
| + | \operatorname{ch}(V)&=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w\cdot \lambda}}{\sum_{w\in W} (-1)^{\ell(w)}e^{w\cdot 0}} \\ | ||
| + | &=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w(\lambda+\rho)}}{\sum_{w\in W} (-1)^{\ell(w)}e^{w(\rho)}}\\ | ||
| + | &=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w(\lambda+\rho)} }{e^{\rho}\prod_{\alpha\in \Delta_+}(1-e^{-\alpha})} | ||
| + | \end{align} | ||
| + | $$ | ||
| + | |||
| + | |||
| + | ===Weyl groups=== | ||
| + | * simple Lie algebras gives the Weyl groups | ||
| + | * for example, the Weyl group associated to $A_2$ is | ||
| + | $$ | ||
| + | \left\langle r_1,r_2 \mid r_1^2=r_2^2=(r_1r_2)^{3}=1\right\rangle | ||
| + | $$ | ||
| + | * there is a class of groups generated by reflections, called Coxeter groups | ||
2016년 7월 18일 (월) 18:12 판
개요
- Bourbaki ordering
- Cartan matrix
- representation of basic objects
- how to represent an element of the root lattice
- how to represent an element of the weight lattice
- how to represent an element of the Weyl group
- change of coordinates from root basis to weight basis and vice versa
- inverse of Cartan matrix
- action of Weyl group on root lattice
- action of Weyl group on weight lattice
- how to generate all positive roots
- how to generate elements of the Weyl group
background
simple Lie algebras
- Lie algebra : vector space with a bilinear, alternating product
$$ [\,,\,]: \mathfrak{g}\times \mathfrak{g} \to \mathfrak{g} $$ satisfying the Jacobi identity \[[a, [b,c]]+[b,[c,a]]+[c,[a,b]]=0\]
- $\mathfrak{sl}_2$ : $2\times 2$ matrix with trace 0 over $\mathbb{C}$ with commutator $[a,b]=ab-ba$
- basis \(\langle e,f,h \rangle\)
\[e=\begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}, f=\begin{pmatrix} 0&0\\ 1&0 \end{pmatrix}, h=\begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}\] \[[h,e]=2e, [h,f]=-2f,[e,f]=h\]
Cartan-Killing
- classification of finite-dim'l simple Lie algebras over $\mathbb{C}$
- key idea : use linear algebra via adjoint representation
- decomposition of $\mathfrak{g}$ relative to a maximal abelian subalgebra $\mathfrak{h}$ -> root space decomposition
- key structure : root systems (highly constrained combinatorial object), $A_2$ example
- possible root system of a simple Lie algebra : $A_l,B_l,C_l,D_l,E_6,E_7,E_8,F_4,G_2$
- this can be compactly encoded in Cartan matrix or Dynkin diagram
Cartan-Weyl
- classification of finite-dim'l irr. rep'n
- key concept : weight space decomposition of rep'n
- Cartan : dominant integral highest weight - finite-dim'l irr. rep'n (weights in the fundamental chamber, $A_2$)
- character of a representaion : generating function of dimension of each weight space
$$\operatorname{ch}(V):=\sum_{\mu \in \mathfrak{h}^{*}} (\dim{V_{\mu}})e^{\mu}$$
- Weyl : character formula, of irr. rep'n \(V=L(\lambda)\) with highest weight $\lambda$
$$ \begin{align} \operatorname{ch}(V)&=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w\cdot \lambda}}{\sum_{w\in W} (-1)^{\ell(w)}e^{w\cdot 0}} \\ &=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w(\lambda+\rho)}}{\sum_{w\in W} (-1)^{\ell(w)}e^{w(\rho)}}\\ &=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w(\lambda+\rho)} }{e^{\rho}\prod_{\alpha\in \Delta_+}(1-e^{-\alpha})} \end{align} $$
Weyl groups
- simple Lie algebras gives the Weyl groups
- for example, the Weyl group associated to $A_2$ is
$$ \left\langle r_1,r_2 \mid r_1^2=r_2^2=(r_1r_2)^{3}=1\right\rangle $$
- there is a class of groups generated by reflections, called Coxeter groups
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