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 \(b_{ij}=a_{ji}\).