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By Miguel A. Rodríguez

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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 ∈ ∆. Este conjunto est´ a formado por: λ − qα, λ − (q − 1)α, . . 75) Entonces v genera un g(α) -m´ odulo irreducible. Los pesos del conjunto definido anteriormente son pesos de h en este subm´ odulo.

Algebras de Lie. Versi´ on 4. 18) En ciertos casos esta expresiones se ven muy simplificadas. Por ejemplo, cuando se toma el cuerpo K. 19) La acci´ on de la aplicaci´ on δ es ahora: m m+1 (−1)i+j ϕ([xi , xj ], x1 , . . , xˆi , . . , xˆj , . . , xm+1 ) δϕ(x1 , . . 23) Por tanto, si el ´ algebra es semisimple (lo que implica [g, g] = g), los 1-cociclos son triviales y el primer grupo de cohomolog´ıa es trivial ({0}). 25) que es la condici´ on que debe cumplir una aplicaci´on bilineal antisim´etrica de g × g en K para ser un 2-cociclo.

103) pues el ´algebra de Virasoro es perfecta. 5. 105) donde c es una base de c. Si rad ψ = {x ∈ g : ψ(x, y) = 0, ∀y ∈ g}, el subespacio c est´a contenido en el radical. Sea ψ˜ la forma inducida en el espacio cociente g/c. 104) y tal que ψ es no degenerada. 1 Consideremos el ´ algebra de polinomios s = K[xj ]j∈J , en la que definimos los operadores: l(xj ), multiplicaci´ on por xj , y ∂/∂xj , derivada parcial con respecto a xj , adem´as del operador identidad, 1. 106) ∂xj es un ´algebra de Heisenberg de dimensi´on, 2 card J + 1.

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Álgebras de Lie by Miguel A. Rodríguez


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