Using the Hopf fibration and starting from a four dimensional noncommutative Moyal plane, R 2 θ × R 2 θ , we obtain a star-product for the noncommutative (fuzzy) R 3Furthermore, we show that there is a projection function which allows us to reduce the functions on R 3 λ to that of the fuzzy sphere, and hence we introduce a new star-product on the fuzzy sphere. We will then briefly discuss how using our method one can extract information about the field theory on fuzzy sphere and R 3 λ from the corresponding field theories on R 2 θ × R 2 θ space.

We consider a gauge theory by taking real quantum groups of nondegenerate bilinear form as a symmetry. The construction of this quantum gauge theory is developed in order to fit with the Hopf algebra structure. In this framework, we show that an appropriate definition of the infinitesimal gauge variations and the axioms of the Hopf algebra structure of the symmetry group lead to the closure of the infinitesimal gauge transformations without any assumption on the commutation rules of the gauge parameters, the connection and the curvature. An adequate definition of the quantum trace is given leading to the quantum Killing form. This is used to construct an invariant quantum Yang–Mills Lagrangian.

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.

: The quantum analogues of Pauli matrices are introduced and investigated. From these matrices and an appropriate trace over spinorial indices we construct a quantum Minkowsky metric. In this framwork we show explicitly the correspondence between the SL(2, C) and Lorentz quantum groups. The R matrices of the quantum Lorentz group are constructed in terms of the R matrices of SL(2, C) group. These R matrices satisfy adequate properties as Yang-Baxter equations, Hecke relations and quantum symmetrization of the metric. It is also shown that the Minkowsky metric leads to an invariant and central norm.