# 유한바일군의 계산 강의노트

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## 개요

• 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 basis of simple roots to basis of fundamental weights 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

### 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 system $\Delta$ (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 $\lambda$

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

### Serre

• Serre 1966 (upon the work of Chevalley, Harish-Chandra, Jacobson)
• Chevalley generators $$e_i,f_i, h_i\, (i=1,\cdots,l)$$
• $$\left[h_i,h_j\right]=0$$
• $$\left[h_i,e_j\right]=a_{ij}e_j$$
• $$\left[h_i,f_j\right]=-a_{ij}f_j$$
• $$\left[e_i,f_j\right]=\delta _{i,j}h_i$$
• $$\left(\operatorname{ad} e_i\right)^{1-a_{ij}}\left(e_j\right)=0$$ ($$i\neq j$$)
• $$\left(\operatorname{ad} f_i\right)^{1-a_{ij}}\left(f_j\right)=0$$ ($$i\neq j$$)
• this defines a simple Lie algebra with Cartan matrix $A$ and settles the existence side of the Cartan-Killing classification project

## Weyl group

### notation

• fix a Cartan matrix $A=(a_{ij})_{i,j\in I}$ of a simple Lie algebra $A_l,B_l,C_l,D_l,E_6,E_7,E_8,F_4,G_2$
• $$P^{\vee} : =\bigoplus_{i\in I}\mathbb{Z}h_{i}$$ : dual weight lattice
• $$\mathfrak{h}: =\mathbb{Q}\otimes_{\mathbb{Z}} P^{\vee}$$
• $$P: =\{\lambda\in\mathfrak{h}^{*}|\lambda(P^{\vee})\subset \mathbb{Z}\}$$ : weight lattice
• $$\Pi^{\vee}:=\{h_{i}\in\mathfrak{h}|i\in I\}$$ : simple coroots
• $$\Pi:=\{\alpha_{i}\in\mathfrak{h}^{*}|i\in I, \alpha_{i}(h_j)=a_{ji}\}$$ : simple roots
• define fundamental weights $\omega_i\in \mathfrak{h}^*$ as $\omega_i(h_j)=\delta_{ij}$ where $\delta_{ij}$ denotes the Kronecker delta
• root lattice $Q= \bigoplus_{i\in I}\mathbb{Z}\alpha_{i}$
• weight lattice $P= \bigoplus_{i\in I}\mathbb{Z}\omega_{i}$

### definition

• define $s_1,\cdots, s_l \in \rm{Aut}(\mathfrak{h}^*)$ by

$$s_i(\lambda) : = \lambda - \lambda(h_i)\alpha_i,\, \lambda\in \mathfrak{h}^*$$

• the Weyl group $W$ is a subgroup of $\rm{Aut}(\mathfrak{h}^*)$ generated by $s_i$
• Explicitly,

$$s_i \omega_j=\omega_j -\delta_{ij}\alpha_i$$

• Note that if $\alpha_i=\sum_{j\in I}b_{ij} \omega_j$, then $b_{ij}=a_{ji}$.