유한바일군의 계산 강의노트
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 $b_{ij}=a_{ji}$.