We present a detailed analysis of the Hamiltonian constraints of the d-dimensional tetrad-connection gravity where the non-dynamic part of the spatial connection is fixed to zero by an adequate gauge transformation. This new action leads to a coherent Hamiltonian formalism where the Lorentz, scalar and vectorial first-class constraints obeying a closed algebra in terms of Poisson brackets. This algebra closes with structure constants instead of structure functions resulting from the Hamiltonian formalisms based on the A.D.M. decomposition. The same algebra of the reduced first-class constraints, where the second-class constraints are eliminated as strong equalities, is obtained in terms of Dirac brackets. These first-class constraints lead to the same physical degrees of freedom of the general relativity.

We construct the vector space dual to the space of right-invariant differential forms constructed from a first-order differential calculus on inhomogeneous quantum groups. We show that this vector space is equipped with a structure of a Hopf algebra which closes on a noncommutative Lie algebra satisfying a Jacobi identity.

From the bicovariant first-order differential calculus on inhomogeneous Hopf algebra ℬ we construct the set of right-invariant Maurer–Cartan one-forms considered as a right-invariant basis of a bicovariant ℬ-bimodule over which we develop the Woronowicz general theory of differential calculus on quantum groups. In this formalism, we introduce suitable functionals on ℬ which control the inhomogeneous commutation rules. In particular, we find that the homogeneous part of commutation rules between the translations and those between the generators of the homogeneous part of ℬ and translations are controlled by different R-matrices satisfying characteristic equations.