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We introduce a family of matrix dilogarithms, which are automorphisms of C^Ntensor C^N, N being any odd positive integer, associated to hyperbolic idealtetrahedra equipped with an additional decoration. The matrix dilogarithmssatisfy fundamental five-term identities that correspond to decorated versionsof the 2 --> 3 move on 3-dimensional triangulations. Together with thedecoration, they arise from the solution we give of a symmetrization problemfor a specific family of basic matrix dilogarithms, the classical (N=1) onebeing the Rogers dilogarithm, which only satisfy one special instance offive-term identity. We use the matrix dilogarithms to construct invariant statesums for closed oriented 3-manifolds $W$ endowed with a flat principalPSL(2,C)-bundle rho, and a fixed non empty link L if N>1, and for (possibly"marked") cusped hyperbolic 3-manifolds M. When N=1 the state sums recoverknown simplicial formulas for the volume and the Chern-Simons invariant. WhenN>$, the invariants for M are new; those for triples (W,L,rho) coincide withthe quantum hyperbolic invariants defined in [Topology 43 (2004) 1373-1423],though our present approach clarifies substantially their nature. We analysethe structural coincidences versus discrepancies between the cases N=1 and N>1,and we formulate "Volume Conjectures", having geometric motivations, about theasymptotic behaviour of the invariants when N tends to infinity.

Classical and quantum dilogarithmic invariants of flatPSL(2,ℂ)–bundles over 3–manifolds