"유한바일군의 계산 강의노트"의 두 판 사이의 차이

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1번째 줄: 1번째 줄:
 
==개요==
 
==개요==
−
* Bourbaki ordering
 
 
* Cartan matrix
 
* Cartan matrix
 
* representation of basic objects
 
* representation of basic objects
6번째 줄: 5번째 줄:
 
** how to represent an element of the weight lattice
 
** how to represent an element of the weight lattice
 
** how to represent an element of the Weyl group
 
** how to represent an element of the Weyl group
−
* change of coordinates from root basis to weight basis and vice versa
+
* change of coordinates from basis of simple roots to basis of fundamental weights and vice versa
 
** inverse of Cartan matrix
 
** inverse of Cartan matrix
 
* action of Weyl group on root lattice
 
* action of Weyl group on root lattice
12번째 줄: 11번째 줄:
 
* how to generate all positive roots
 
* how to generate all positive roots
 
* how to generate elements of the Weyl group
 
* how to generate elements of the Weyl group
−
 
  
 
==background==
 
==background==
−
===simple Lie algebras===
+
===Lie algebras===
 
* Lie algebra : vector space with a bilinear, alternating product
 
* Lie algebra : vector space with a bilinear, alternating product
−
$$
+
:<math>
 
[\,,\,]: \mathfrak{g}\times \mathfrak{g} \to \mathfrak{g}
 
[\,,\,]: \mathfrak{g}\times \mathfrak{g} \to \mathfrak{g}
−
$$
+
</math>
 
satisfying the Jacobi identity
 
satisfying the Jacobi identity
 
:<math>[a, [b,c]]+[b,[c,a]]+[c,[a,b]]=0</math>
 
:<math>[a, [b,c]]+[b,[c,a]]+[c,[a,b]]=0</math>
−
* $\mathfrak{sl}_2$ : $2\times 2$ matrix with trace 0 over $\mathbb{C}$ with commutator $[a,b]=ab-ba$
+
* <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>
 
* 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>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>
 
:<math>[h,e]=2e, [h,f]=-2f,[e,f]=h</math>
 +
 
===Cartan-Killing===
 
===Cartan-Killing===
−
* classification of finite-dim'l simple Lie algebras over $\mathbb{C}$
+
* classification of finite-dim'l simple Lie algebras over <math>\mathbb{C}</math>
 
* key idea : use linear algebra via adjoint representation
 
* key idea : use linear algebra via adjoint representation
−
* decomposition of $\mathfrak{g}$ relative to a maximal abelian subalgebra $\mathfrak{h}$ -> root space decomposition
+
* decomposition of <math>\mathfrak{g}</math> relative to a maximal abelian subalgebra <math>\mathfrak{h}</math> -> root space decomposition
−
* key structure : root systems (highly constrained combinatorial object), $A_2$ example
+
* key structure : root system <math>\Delta</math> (highly constrained combinatorial object), <math>A_2</math> example
−
* possible root system of a simple Lie algebra : $A_l,B_l,C_l,D_l,E_6,E_7,E_8,F_4,G_2$
+
* 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
 
* this can be compactly encoded in Cartan matrix or Dynkin diagram
  
37번째 줄: 36번째 줄:
 
* classification of finite-dim'l irr. rep'n  
 
* classification of finite-dim'l irr. rep'n  
 
* key concept : weight space decomposition of 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, $A_2$)
+
* 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
 
* character of a representaion : generating function of dimension of each weight space
−
$$\operatorname{ch}(V):=\sum_{\mu \in \mathfrak{h}^{*}} (\dim{V_{\mu}})e^{\mu}$$
+
:<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 $\lambda$
+
* Weyl : character formula,  of irr. rep'n <math>V=L(\lambda)</math> with highest weight <math>\lambda</math>
−
$$
+
:<math>
−
\begin{align}
+
\operatorname{ch}(V)=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w(\lambda+\rho)} }{e^{\rho}\prod_{\alpha\in \Delta_+}(1-e^{-\alpha})}
−
\operatorname{ch}(V)&=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w\cdot \lambda}}{\sum_{w\in W} (-1)^{\ell(w)}e^{w\cdot 0}} \\
+
</math>
−
&=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w(\lambda+\rho)}}{\sum_{w\in W} (-1)^{\ell(w)}e^{w(\rho)}}\\
+
 
−
&=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w(\lambda+\rho)} }{e^{\rho}\prod_{\alpha\in \Delta_+}(1-e^{-\alpha})}
+
===Serre===
−
\end{align}
+
* 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 groups===
+
==Weyl group==
−
* simple Lie algebras gives the Weyl groups
+
===notation===
−
* for example, the Weyl group associated to $A_2$ is
+
* 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>
−
$$
+
* <math>P^{\vee} : =\bigoplus_{i\in I}\mathbb{Z}h_{i}</math> : dual weight lattice
−
\left\langle r_1,r_2 \mid r_1^2=r_2^2=(r_1r_2)^{3}=1\right\rangle
+
* <math>\mathfrak{h}: =\mathbb{Q}\otimes_{\mathbb{Z}} P^{\vee}</math>
−
$$
+
* <math>P: =\{\lambda\in\mathfrak{h}^{*}|\lambda(P^{\vee})\subset \mathbb{Z}\}</math> : weight lattice
−
* there is a class of groups generated by reflections, called Coxeter groups
+
* <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>.
  
 
==관련된 항목들==
 
==관련된 항목들==
 +
* [[바일 지표 공식 (Weyl character formula)]]
 
* [[코스탄트 무게 중복도 공식 (Kostant weight multiplicity formula)]]
 
* [[코스탄트 무게 중복도 공식 (Kostant weight multiplicity formula)]]
−
 
  
 
==매스매티카 파일==
 
==매스매티카 파일==

2020년 11월 16일 (월) 04:24 기준 최신판

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

관련된 항목들

매스매티카 파일

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