Álgebras de Lie by Miguel A. Rodríguez

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By Miguel A. Rodríguez

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Adem´ as adx (N ) ⊂ k, luego adx act´ ua trivialmente en el cociente. Esto solo puede ser si el cociente es trivial, es decir, si k = N (k). Veamos algunas propiedades elementales de las sub´algebras de Cartan. 4 Sea g un ´ algebra de Lie de dimensi´ on finita y h una sub´ algebra nilpotente de g. adh adh Entonces, h es una sub´ algebra de Cartan si y solo si h = g0 donde g0 es la componente nula de la descomposici´ on de Fitting de g respecto a ad h (es decir el subespacio propio generalizado de autovalor 0 de ad h) (ver [4], p´ agina 80).

Versi´on 4. 3. 49) y sea π una representaci´ on de dimensi´ on finita en un espacio V . Sean H = π(h), E = π(e), F = π(f ). Los endomorfismos {H, E, F } verifican las mismas relaciones de conmutaci´on anteriores. 51) De esta manera obtenemos una sucesi´ on de vectores en V linealmente independientes (porque corresponden a distintos autovalores de H) y que necesariamente debe acabar, pues V es de dimensi´on finita. 52) Sea v = E p v. 54) Aplicamos ahora el operador F . Tenemos una sucesi´on de autovectores v0 = v, v1 = F v, .

17) ˜ = (adx )l g. Entonces: Sea g0 = χ(ad ˜ x )g, g ˜ (como espacios vectoriales, se trata de la descomposici´ 1. g = g0 ⊕ g on de Fitting) 2. g0 es una sub´ algebra de Cartan, dim g0 = l ˜] = [x, g ˜] = g ˜ 3. [g0 , g Veamos la relaci´ on entre los espacios de ra´ıces. Sea g un ´algebra de Lie de dimensi´on finita, h una sub´algebra nilpotente y V un m´odulo de g, con representaci´on π. Supongamos (como ya hemos dicho) que el cuerpo K es algebraicamente cerrado (o que los polinomios cracter´ısticos involucrados tienen todas sus ra´ıces en el cuerpo).

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