Tabla de productos para el producto interior de raíces y pesos del álgebra de Lie

Question

Para un álgebra de Lie simple $\frak{g}$ es habitual escalar el producto interior para que la raíz simple más corta tenga longitud $2$ . Con estas convenciones, ¿dónde puedo encontrar una tabla (en línea) de los siguientes valores para un simple general $\frak{g}$ :

i) $(\alpha_i,\alpha_j)$ para todos $i,j=1,\dots, r:=\mathrm{rank}(\frak{g})$

ii) $(\alpha_i,\pi_j)$ , para $\pi_j$ un peso fundamental

iii) $(\pi_i,\pi_j)$ ?

Answer 1

Esta información está disponible mediante el software Atlas of Lie Groups and Representations http://www.liegroups.org/software . Este es un ejemplo, los comentarios están entre llaves.

atlas> set G=Sp(4)
Variable G: RootDatum
atlas> set f=G.invariant_form {W-invariant bilinear form}
Added definition [3] of f: (ratvec,ratvec->rat)
atlas> set alpha=highest_short_root(G)
Variable alpha: vec
atlas> f(alpha,alpha)
Value: 6/1  {need to normalize it differently}
atlas> set g(ratvec v,ratvec w)=f(v,w)/3
Added definition [3] of g: (ratvec,ratvec->rat)
atlas> g(alpha,alpha)  {normalized invariant form}
Value: 2/1
atlas> void:for alpha in G.posroots do 
for beta in G.posroots do
prints(alpha, " ", beta, " ", g(alpha,beta)) od od
[  1, -1 ] [  1, -1 ] 2/1
[  1, -1 ] [ 0, 2 ] -2/1
[  1, -1 ] [ 1, 1 ] 0/1
[  1, -1 ] [ 2, 0 ] 2/1
[ 0, 2 ] [  1, -1 ] -2/1
[ 0, 2 ] [ 0, 2 ] 4/1
[ 0, 2 ] [ 1, 1 ] 2/1
[ 0, 2 ] [ 2, 0 ] 0/1
[ 1, 1 ] [  1, -1 ] 0/1
[ 1, 1 ] [ 0, 2 ] 2/1
[ 1, 1 ] [ 1, 1 ] 2/1
[ 1, 1 ] [ 2, 0 ] 2/1
[ 2, 0 ] [  1, -1 ] 2/1
[ 2, 0 ] [ 0, 2 ] 0/1
[ 2, 0 ] [ 1, 1 ] 2/1
[ 2, 0 ] [ 2, 0 ] 4/1
atlas> void:for alpha in G.posroots do for beta in
G.fundamental_weights do 
prints(alpha, " ", beta, " ", g(alpha,beta)) od od
[  1, -1 ] [ 1, 0 ]/1 1/1
[  1, -1 ] [ 1, 1 ]/1 0/1
[ 0, 2 ] [ 1, 0 ]/1 0/1
[ 0, 2 ] [ 1, 1 ]/1 2/1
[ 1, 1 ] [ 1, 0 ]/1 1/1
[ 1, 1 ] [ 1, 1 ]/1 2/1
[ 2, 0 ] [ 1, 0 ]/1 2/1
[ 2, 0 ] [ 1, 1 ]/1 2/1
atlas> void:for alpha in G.fundamental_weights do
for beta in G.fundamental_weights do 
prints(alpha, " ", beta, " ", g(alpha,beta)) od od
[ 1, 0 ]/1 [ 1, 0 ]/1 1/1
[ 1, 0 ]/1 [ 1, 1 ]/1 1/1
[ 1, 1 ]/1 [ 1, 0 ]/1 1/1
[ 1, 1 ]/1 [ 1, 1 ]/1 2/1