유한바일군의 계산 강의노트
둘러보기로 가기
검색하러 가기
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
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 \(b_{ij}=a_{ji}\).