Classical Lie algebras

1 Classical Lie algebras A Lie algebra is a vector space g with a bilinear map [, ] : g× g → g such that (a) [x, y] = −[y, x], for x, y ∈ g, and (b) (Jacobi identity) [x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0, for all x, y, z ∈ g. A bilinear form 〈, 〉 : g× g → C is ad-invariant if, for all x, y, z ∈ g, 〈adx(y), z〉 = −〈y, adx(z)〉, where adx(y) = [x, y], (1.1) for x, y,∈ g. The Killing form is the inner product on g given by 〈x1, x2〉 = Tr(adxady)〉. (1.2) The Jacobi identity is equivalent to the fact that the Killing form is ad-invariant. Let g be a finite dimensional Lie algebra with a nondegenerate ad-invariant bilinear form. The nondegeneracy of the form means that if {xi} be a basis of g then the dual basis {xi } of g with respect to 〈, 〉 exists. The Casimir element of g is κ = ∑