유한바일군의 계산 강의노트
개요
- Bourbaki ordering
- 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 root basis to weight basis 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
simple 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 $\lambda$
$$ \operatorname{ch}(V)=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w(\lambda+\rho)} }{e^{\rho}\prod_{\alpha\in \Delta_+}(1-e^{-\alpha})} $$
Weyl groups
- simple Lie algebras gives the Weyl groups
- for example, the Weyl group associated to $A_2$ is
$$ \left\langle r_1,r_2 \mid r_1^2=r_2^2=(r_1r_2)^{3}=1\right\rangle $$
- there is a class of groups generated by reflections, called Coxeter groups
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