The lagrangian of q-Poincar e ́ gravity

The gauging of the q-Poincar e ́ algebra of L. Castellani [Differential calculus on ISO_q(N), quantum Poincaré algebra and q-gravity, Torino preprint DFTT-70/93, hep-th 9312179] yields a non-commutative generalization of the Einstein-Cartan lagrangian. We prove its invariance under local q-Lorentz rotations and, up to a total derivative, under q-diffeomorphisms. The variations of the fields are given by their q-Lie derivative, in analogy with the q = 1 case. The algebra of q-Lie derivatives is shown to close with field dependent structure functions. The equations of motion are found, generalizing the Einstein equations and the zero-torsion condition.