"유한바일군의 계산 강의노트"의 두 판 사이의 차이

2016년 7월 18일 (월) 19:15 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 $\lambda$

$$ \operatorname{ch}(V)=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w(\lambda+\rho)} }{e^{\rho}\prod_{\alpha\in \Delta_+}(1-e^{-\alpha})} $$

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

매스매티카 파일

https://drive.google.com/file/d/0B8XXo8Tve1cxTG5GeDQtRS1fMDA/view

@@ 42번째 줄: / 42번째 줄: @@
 * Weyl : character formula,  of irr. rep'n <math>V=L(\lambda)</math> with highest weight $\lambda$
 $$
-\begin{align}
+\operatorname{ch}(V)=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w(\lambda+\rho)} }{e^{\rho}\prod_{\alpha\in \Delta_+}(1-e^{-\alpha})}
-\operatorname{ch}(V)&=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w\cdot \lambda}}{\sum_{w\in W} (-1)^{\ell(w)}e^{w\cdot 0}} \\
-&=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w(\lambda+\rho)}}{\sum_{w\in W} (-1)^{\ell(w)}e^{w(\rho)}}\\
-&=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w(\lambda+\rho)} }{e^{\rho}\prod_{\alpha\in \Delta_+}(1-e^{-\alpha})}
-\end{align}
 $$
 ===Weyl groups===