Dual formulation of the Lie algebra S-expansion procedure

The expansion of a Lie algebra entails finding a new bigger algebra [fraktur G] through a series of well-defined steps from an original Lie algebra [fraktur 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.” ©2009 American Institute of Physic