Dual formulation of the Lie algebra S-expansion procedure

The expansion of a Lie algebra entails finding a new bigger algebra G through a series of well-defined steps from an original Lie algebra g. One incarnation of the method, the so-called S-expansion, involves the use of a finite Abelian semigroup S to accomplish this task. In this paper we put forward a dual formulation of the S-expansion method, which is based on the dual picture of a Lie algebra given by the Maurer-Cartan forms. The dual version of the method is useful in finding a generalization to the case of a gauge free differential algebra, which, in turn, is relevant for physical applications in, e.g., supergravity. It also sheds new light on the puzzling relation between two Chern-Simons Lagrangians for gravity in 2+1 dimensions, namely, the Einstein-Hilbert Lagrangian and the one for the so-called exotic gravity..