유한바일군의 계산 강의노트

수학노트
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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 $b_{ij}=a_{ji}$.

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