Generating Lie and gauge free differential (super)algebras by expanding Maurer–Cartan forms and Chern–Simons supergravity

Abstract

We study how to generate new Lie algebras G(N 0,…,N p,…,N n) from a given one G . The (order by order) method consists in expanding its Maurer–Cartan one-forms in powers of a real parameter λ which rescales the coordinates of the Lie (super)group G, g i p → λ p g i p , in a way subordinated to the splitting of G as a sum V 0⊕⋯⊕ V p ⊕⋯⊕ V n of vector subspaces. We also show that, under certain conditions, one of the obtained algebras may correspond to a generalized İnönü–Wigner contraction in the sense of Weimar-Woods, but not in general. The method is used to derive the M-theory superalgebra, including its Lorentz part, from osp(1|32). It is also extended to include gauge free differential (super)algebras and Chern–Simons theories, and then applied to D=3 CS supergravity.