We investigate the representation theory of the polynomial core T q S of the quantum Teichmüller space of a punctured surface S . This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of ideal cell decompositions of S . Our main result is that irreducible finite-dimensional representations of T q S are classified, up to finitely many choices, by group homomorphisms from the fundamental group 1 .S/ to the isometry group of the hyperbolic 3-space ‫ވ‬ 3 . We exploit this connection between algebra and hyperbolic geometry to exhibit invariants of diffeomorphisms of S . 57R56; 57M50, 20G42This work finds its motivation in the emergence of various conjectural connections between topological quantum field theory and hyperbolic geometry, such as the now famous Volume Conjecture of Rinat Kashaev [22], and Hitochi Murakami and Jun Murakami [28]. For a hyperbolic link L in the 3-sphere S 3 , this conjectures relates the hyperbolic volume of the complement S 3 L to the asymptotic behavior of the N -th colored Jones polynomial J N L .e 2 i=N / of L, evaluated at the primitive N -th root of unity e 2 i=N . At this point, the heuristic evidence (Kashaev [22], Murakami, Murakami, Okamoto, Takata and Yokota [29], and Yokota [43; 44]) for the Volume Conjecture is based on the observation [22; 28] that the N -th Jones polynomial can be computed using an explicit R-matrix whose asymptotic behavior is related to Euler's dilogarithm function, which is well-known to give the hyperbolic volume of an ideal tetrahedron in ‫ވ‬ 3 in terms of the cross-ratio of its vertices. We wanted to establish a more conceptual connection between the two points of view, namely between quantum algebra and 3-dimensional hyperbolic geometry.We investigate such a relationship, provided by the quantization of the Teichmüller space of a surface, as developed by Rinat Kashaev [23], Leonid Chekhov and Vladimir Fock [12]. More precisely, we follow the exponential version of the Chekhov-Fock approach. This enables us to formulate our discussion in terms of non-commutative algebraic geometry and finite-dimensional representations of algebras, instead of Lie algebras and self-adjoint operators of Hilbert spaces. This may be physically less More precisely, let S be a surface of finite topological type, with genus g and with p > 1 punctures. An ideal triangulation of S is a proper 1-dimensional submanifold whose complementary regions are infinite triangles with vertices at infinity, namely at the punctures. For an ideal triangulation and a number q D e i" 2 ‫,ރ‬ the ChekhovFock algebra T q is the algebra over ‫ރ‬ defined by generators X˙1 1 , X˙1 2 , . . . , X˙1 n associated to the components of and by relations X i X j D q 2 ij X j X i , where the ij are integers determined by the combinatorics of the ideal triangulation . This algebra has a well-defined fraction division algebra y T q . In concrete terms, T q consists of the formal Laurent polynomials in variables X i satisfying the skew-commutativity relations This con...

The present investigation discusses the advantage of using RE-ion-doped (Nd 3؉ , Tm 3؉ , and Er 3؉ ) TeO 2 glasses for developing fiber and planar broadband amplifiers and lasers. The spectroscopy of RE-ion-doped fibers and glasses is discussed along with the thermal properties of glass hosts. The results of emission from the 3 H 4 level in single-mode Tm 3؉doped tellurite fiber show that the emission band overlaps with Er 3؉ emission from the 4 I 13/2 level and Nd 3؉ emission from the 4 F 3/2 level in silicate and tellurite glasses, thereby enabling the development of amplifiers and lasers between 1350 and 1650 nm. Recent results using Z-scan measurements of nonlinear refractive index and absorption demonstrate that the thirdorder nonlinearity in undoped TeO 2 glasses is of the order of 2 ؋ 10 ؊15 to 3 ؋ 10 ؊15 cm 2 ⅐W ؊1 between 1300 and 1550 nm. These results are briefly discussed in view of an amplifier operation combined with ultrafast all-optical switching.

The visible and near infrared emission spectra of Er(3+)-Tm3(3+)-codoped tellurite glasses and fibres were measured with the excitation of an 800 nm laser. A broad emission extending from 1.35 mum to 1.6 mum with a full width at half-maximum (FWHM) of ~160 nm was recorded in a 24 cm long 0.2 wt% Er(2)O(3) and 1.0 wt% Tm(2)O(3) codoped tellurite fibre. Energy transfer between Er(3+) and Tm(3+) play important roles in the luminescence mechanism. These results indicate that Er(3+)-Tm(3+) codoped tellurite fibre could be a promising material for broadband light source and broadband amplifier for the wavelength-division-multiplexing (WDM) transmission systems.

No abstract