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
- <math>
[\,,\,]: \mathfrak{g}\times \mathfrak{g} \to \mathfrak{g} </math> satisfying the Jacobi identity
- <math>[a, [b,c]]+[b,[c,a]]+[c,[a,b]]=0</math>
- <math>\mathfrak{sl}_2</math> : <math>2\times 2</math> matrix with trace 0 over <math>\mathbb{C}</math> with commutator <math>[a,b]=ab-ba</math>
- basis <math>\langle e,f,h \rangle</math>
- <math>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}</math>
- <math>[h,e]=2e, [h,f]=-2f,[e,f]=h</math>
Cartan-Killing
- classification of finite-dim'l simple Lie algebras over <math>\mathbb{C}</math>
- key idea : use linear algebra via adjoint representation
- decomposition of <math>\mathfrak{g}</math> relative to a maximal abelian subalgebra <math>\mathfrak{h}</math> -> root space decomposition
- key structure : root system <math>\Delta</math> (highly constrained combinatorial object), <math>A_2</math> example
- possible root system of a simple Lie algebra : <math>A_l,B_l,C_l,D_l,E_6,E_7,E_8,F_4,G_2</math>
- 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, <math>A_2</math>)
- character of a representaion : generating function of dimension of each weight space
- <math>\operatorname{ch}(V):=\sum_{\mu \in \mathfrak{h}^{*}} (\dim{V_{\mu}})e^{\mu}</math>
- Weyl : character formula, of irr. rep'n <math>V=L(\lambda)</math> with highest weight <math>\lambda</math>
- <math>
\operatorname{ch}(V)=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w(\lambda+\rho)} }{e^{\rho}\prod_{\alpha\in \Delta_+}(1-e^{-\alpha})} </math>
Serre
- Serre 1966 (upon the work of Chevalley, Harish-Chandra, Jacobson)
- Chevalley generators <math>e_i,f_i, h_i\, (i=1,\cdots,l)</math>
- <math>\left[h_i,h_j\right]=0</math>
- <math>\left[h_i,e_j\right]=a_{ij}e_j</math>
- <math>\left[h_i,f_j\right]=-a_{ij}f_j</math>
- <math>\left[e_i,f_j\right]=\delta _{i,j}h_i</math>
- <math>\left(\operatorname{ad} e_i\right)^{1-a_{ij}}\left(e_j\right)=0</math> (<math>i\neq j</math>)
- <math>\left(\operatorname{ad} f_i\right)^{1-a_{ij}}\left(f_j\right)=0</math> (<math>i\neq j</math>)
- this defines a simple Lie algebra with Cartan matrix <math>A</math> and settles the existence side of the Cartan-Killing classification project
Weyl group
notation
- fix a Cartan matrix <math>A=(a_{ij})_{i,j\in I}</math> of a simple Lie algebra <math>A_l,B_l,C_l,D_l,E_6,E_7,E_8,F_4,G_2</math>
- <math>P^{\vee} : =\bigoplus_{i\in I}\mathbb{Z}h_{i}</math> : dual weight lattice
- <math>\mathfrak{h}: =\mathbb{Q}\otimes_{\mathbb{Z}} P^{\vee}</math>
- <math>P: =\{\lambda\in\mathfrak{h}^{*}|\lambda(P^{\vee})\subset \mathbb{Z}\}</math> : weight lattice
- <math>\Pi^{\vee}:=\{h_{i}\in\mathfrak{h}|i\in I\}</math> : simple coroots
- <math>\Pi:=\{\alpha_{i}\in\mathfrak{h}^{*}|i\in I, \alpha_{i}(h_j)=a_{ji}\}</math> : simple roots
- define fundamental weights <math>\omega_i\in \mathfrak{h}^*</math> as <math>\omega_i(h_j)=\delta_{ij}</math> where <math>\delta_{ij}</math> denotes the Kronecker delta
- root lattice <math>Q= \bigoplus_{i\in I}\mathbb{Z}\alpha_{i}</math>
- weight lattice <math>P= \bigoplus_{i\in I}\mathbb{Z}\omega_{i}</math>
definition
- define <math>s_1,\cdots, s_l \in \rm{Aut}(\mathfrak{h}^*)</math> by
- <math>
s_i(\lambda) : = \lambda - \lambda(h_i)\alpha_i,\, \lambda\in \mathfrak{h}^* </math>
- the Weyl group <math>W</math> is a subgroup of <math>\rm{Aut}(\mathfrak{h}^*)</math> generated by <math>s_i</math>
- Explicitly,
- <math>s_i \omega_j=\omega_j -\delta_{ij}\alpha_i</math>
- Note that if <math>\alpha_i=\sum_{j\in I}b_{ij} \omega_j</math>, then <math>b_{ij}=a_{ji}</math>.