The Vanishing of Scalar Curvature on 6 Manifolds, Einstein’s Equation, and Representation Theory
Let G be a real semisimple Lie Group, which we may assume to be connected or even simply connected, and let g0 = Lie(G). The dual vector space of g0 will be denoted by g0*. Also, g and g* will denote the complexifications of g0 and g0*, respectively. We recall that the group G and the Lie algebra g operate on g by the adjoint representation and on g* by the so-called coadjoint representation. Moreover, set G ℂ = Ad(g). Since the bilinear form (x, y) = tr(ad x ady) on g × g is nonsingular, we can identify g and g*, which we shall do whenever it is convenient.
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