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Álgebras de Lie by Miguel A. Rodríguez

By Miguel A. Rodríguez

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Este conjunto es una Q-´algebra graduada. Las graduaciones de ´ algebras son muy importantes como veremos en algunos ejemplos. 3. Extensiones centrales Las extensiones centrales de ´ algebras de Lie aparecen frecuentemente en F´ısica. Por ejemplo cuando se estudia el ´ algebra del grupo de Galileo o las ´algebras de Kac-Moody. Una sucesi´ on exacta de ´ algebras de Lie es un conjunto de ´algebras y homomorfismos: fj−1 fj · · · → gj−1 −→ gj −→ gj+1 → · · · tal que im fj−1 = ker fj . 35) ´ MAR. Algebras de Lie.

110) una forma bilineal, invariante no degenerada. 111) i Entonces, el endomorfismo Cφ (β) conmuta con todos los φ(x). En efecto: [φ(x), Cφ (β)] = [φ(x), φ(xi )φ(yi )] = i [φ(x), φ(xi )]φ(yi ) + i φ(xi )[φ(x), φ(yi )] = (aij + bji )φ(xj )φ(yi ) = i 0 El operador Cφ (β) es un Casimir del ´algebra. El valor de su traza es justamente la dimensi´on del ´algebra. Si φ es una representaci´ on irreducible, el lema de Schur asegura que se trata de un m´ ultiplo de la identidad. El valor de esta constante es la dimensi´on del a´lgebra dividida por la dimensi´on de la representaci´ on.

La expresi´on que debe satisfacer es (x, y, z ∈ a): [β(x), ϕ(y, z)] − [β(y), ϕ(x, z)] + [β(z), ϕ(x, y)] − ϕ([x, y], z) + ϕ([x, z], y) − ϕ([y, z], x) = 0 identidad que se comprueba sin m´ as que sustituir ϕ por su definici´on en funci´on de β. 50) Sustituyendo: [β([x, y]), β(z)] + [β([y, z]), β(x)] + [β([z, x]), β(y)]− β([[x, y], z]) − β([[y, z], x]) − β([[z, x], y]) [β([x, y]), β(z)] + [β([y, z]), β(x)] + [β([z, x]), β(y)] = pues β es lineal y podemos aplicar la propiedad de Jacobi. 3. Veamos ahora que ocurre si cambiamos la secci´on β por otra β .

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