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By Miguel A. Rodríguez

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Daremos su definici´ on y algunas propiedades fundamentales siguiendo a [8]. 71) Se llama a {X, [ ]} el ´ algebra de Witt, W. Es un ´algebra perfecta, [W, W] = W como se puede comprobar f´ acilmente. 73) ´ MAR. Algebras de Lie. Versi´ on 4. 74) una suma de espacios vectoriales, con 2-cociclo α. El resultado fundamental es que el ´algebra de Virasoro es el recubridor universal del ´algebra de Witt. Ve´ amoslo con cierto detalle. 76) Demos algunos valores particulares a k, m, n: 1. 77) 2. 79) Definimos ahora el par´entesis en la extensi´on W ⊕ V de la forma que vimos en la secci´on anterior.

Yn ) = xy1 . . 39) Cap´ıtulo 5 Algunos complementos La teor´ıa de ´ algebras de Lie presenta muchos aspectos que no pueden ser tratados de una manera completa aqu´ı. Nos limitaremos a esbozar algunos rudimentos y ejemplos interesantes desde el punto de vista de las aplicaciones en F´ısica. 1. Cohomolog´ıa de ´ algebras de Lie Veamos la definici´ on y algunas aplicaciones a casos sencillos. Sea g un ´ algebra de Lie y V un g-m´odulo (la acci´on se designar´a por ·). 1) que verifica: 1. ϕ es k-lineal 2.

56 no singular en gα . 35) como quer´ıamos probar. Supongamos ahora que eβ ∈ gβ , con α + β = 0. 36) Puede ocurrir que α + β sea una ra´ız (no cero). Entonces [e1α , eβ ] ∈ gα+β y por lo anterior K(eα , eβ ) = 0. Si α + β no es una ra´ız, entonces [e1α , eβ ] = 0 y se tiene el mismos resultado. 8 La restricci´ on de la forma de Killing a la sub´ algebra de Cartan es no degenerada Si K|h fuera degenerada, existir´ıa h ∈ h tal que K(h, h ) = 0 para todo h ∈ h. Pero como h es ortogonal a cualquier elemento de gα , para toda ra´ız α, se tiene que K ser´ıa degenerada en g lo que no es posible al ser g semisimple.

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