Kac-Moody spectrum of (half-)maximal supergravities

We establish the correspondence between, on one side, the possible gaugings and massive deformations of half-maximal supergravity coupled to vector multiplets and, on the other side, certain generators of the associated very extended Kac-Moody algebras. The difference between generators associated to gaugings and to massive deformations is pointed out. Furthermore, we argue that another set of generators are related to the so-called quadratic constraints of the embedding tensor. Special emphasis is placed on a truncation of the Kac-Moody algebra that is related to the bosonic gauge transformations of supergravity. We give a separate discussion of this truncation when non-zero deformations are present. The new insights are also illustrated in the context of maximal supergravity