New PDF release: Álgebras de Lie

By Miguel A. Rodríguez

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Se define un espacio vectorial como suma de los espacios vectoriales asociados a g y a. 69) y en ´el un par´entesis de Lie dado por: Est´a bien definido debido a las propiedades de cociclo de α. Entonces, g es un ´algebra de Lie y, si se definen la inclusi´ on: i : a → g y la proyecci´on π : g → g se tiene una extensi´on central. 4. El ´ algebra de Virasoro El ´algebra de Virasoro es un objeto fundamental en la teor´ıa de ´algebras de Lie de dimensi´on infinita. Daremos su definici´ on y algunas propiedades fundamentales siguiendo a [8].

Vj } tal que: π(h)vk = (λ − kα)(h)vk , k = 0, 1, . . , j π(e−α )vk = vk+1 , k = 0, 1, . . , j − 1, π(e−α )vj = 0 1 π(eα )vk = k(j − k + 1) α, α vk−1 , k = 1, 2, . . 72) La demostraci´ on se basa en un uso adecuado de los resultados obtenidos para sl(2, C). 17 Si v ∈ V es distinto de cero, tal que π(h)v = λ(h)v, π(eα )v = 0, entonces v genera un g(α) -m´ odulo irreducible. 18 Sea λ un peso de la representaci´ on (V, π). Consideremos el conjunto de pesos que tiene la forma λ + kα, k ∈ Z, α ∈ ∆.

58) como quer´ıamos probar. El espacio Wm es entonces invariante. La traza del operador H (siendo el conmutador de otros dos operadores) debe ser cero. 60) ´ MAR. Algebras de Lie. Versi´ on 4. 59 y j es un entero, as´ı como el resto de los autovalores de H. Probemos finalmente que el subespacio ˜ ⊂ Wj es un subespacio que contiene a un vector de la forma w = ci vi + Wj es irreducible. Si W · · · cj vj , con ci = 0 y es invariante bajo H, E, F entonces contiene a todos los vectores xi y es igual a Wj .

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Álgebras de Lie by Miguel A. Rodríguez

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