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- W2009145422 abstract "We introduce a family of matrix dilogarithms, which are automorphisms of C^N tensor C^N, N being any odd positive integer, associated to hyperbolic ideal tetrahedra equipped with an additional decoration. The matrix dilogarithms satisfy fundamental five-term identities that correspond to decorated versions of the 2 --> 3 move on 3-dimensional triangulations. Together with the decoration, they arise from the solution we give of a symmetrization problem for a specific family of basic matrix dilogarithms, the classical (N=1) one being the Rogers dilogarithm, which only satisfy one special instance of five-term identity. We use the matrix dilogarithms to construct invariant state sums for closed oriented 3-manifolds $W$ endowed with a flat principal PSL(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 recover known simplicial formulas for the volume and the Chern-Simons invariant. When N>$, the invariants for M are new; those for triples (W,L,rho) coincide with the quantum hyperbolic invariants defined in [Topology 43 (2004) 1373-1423], though our present approach clarifies substantially their nature. We analyse the structural coincidences versus discrepancies between the cases N=1 and N>1, and we formulate Volume Conjectures, having geometric motivations, about the asymptotic behaviour of the invariants when N tends to infinity." @default.
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- W2009145422 date "2005-04-08" @default.
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- W2009145422 title "Classical and quantum dilogarithmic invariants of flatPSL(2,ℂ)–bundles over 3–manifolds" @default.
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- W2009145422 doi "https://doi.org/10.2140/gt.2005.9.493" @default.
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