The Vanishing of Scalar Curvature on 6 Manifolds, Einstein’s Equation, and Representation Theory

Abstract

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.

We wish to thank Juan Tirao for his contribution to these notes.