Operator Coproduct-Realization of Quantum Group Transformations in Two Dimensional Gravity, I.

A simple connection between the universal $R$ matrix of $U_q(sl(2))$ (for spins \demi and $J$) and the required form of the co-product action of the Hilbert space generators of the quantum group symmetry is put forward. This gives an explicit operator realization of the co-product action on the covariant operators. It allows us to derive the quantum group covariance of the fusion and braiding matrices, although it is of a new type: the generators depend upon worldsheet variables, and obey a new central extension of $U_q(sl(2))$ realized by (what we call) fixed point commutation relations. This is explained by showing that the link between the algebra of field transformations and that of the co-product generators is much weaker than previously thought. The central charges of our extended $U_q(sl(2))$ algebra, which includes the Liouville zero-mode momentum in a nontrivial way are related to Virasoro-descendants of unity. We also show how our approach can be used to derive the Hopf algebra structure of the extended quantum-group symmetry U_q(sl(2))\odot U_{\qhat}(sl(2)) related to the presence of both of the screening charges of 2D gravity.Comment: 33 pages, latex, no figure

Quantum Group Structure and Local Fields in the Algebraic Approach to 2D Gravity

This review contains a summary of work by J.-L. Gervais and the author on the operator approach to 2d gravity. Special emphasis is placed on the construction of local observables -the Liouville exponentials and the Liouville field itself - and the underlying algebra of chiral vertex operators. The double quantum group structure arising from the presence of two screening charges is discussed and the generalized algebra and field operators are derived. In the last part, we show that our construction gives rise to a natural definition of a quantum tau function, which is a noncommutative version of the classical group-theoretic representation of the Liouville fields by Leznov and Saveliev.Comment: 38 pages, LaTex file. Proceedings of the Vth International Conference on Mathematical Physics, Strings and Quantum gravity, Alushta, Ukraine 199

The Quantum Group Structure of 2D Gravity and Minimal Models II: The Genus-Zero Chiral Bootstrap

The F and B matrices associated with Virasoro null vectors are derived in closed form by making use of the operator-approach suggested by the Liouville theory, where the quantum-group symmetry is explicit. It is found that the entries of the fusing and braiding matrices are not simply equal to quantum-group symbols, but involve additional coupling constants whose derivation is one aim of the present work. Our explicit formulae are new, to our knowledge, in spite of the numerous studies of this problem. The relationship between the quantum-group-invariant (of IRF type) and quantum-group-covariant (of vertex type) chiral operator-algebras is fully clarified, and connected with the transition to the shadow world for quantum-group symbols. The corresponding 3-j-symbol dressing is shown to reduce to the simpler transformation of Babelon and one of the author (J.-L. G.) in a suitable infinite limit defined by analytic continuation. The above two types of operators are found to coincide when applied to states with Liouville momenta going to $\infty$ in a suitable way. The introduction of quantum-group-covariant operators in the three dimensional picture gives a generalisation of the quantum-group version of discrete three-dimensional gravity that includes tetrahedra associated with 3-j symbols and universal R-matrix elements. Altogether the present work gives the concrete realization of Moore and Seiberg's scheme that describes the chiral operator-algebra of two-dimensional gravity and minimal models.Comment: 56 pages, 22 figures. Technical problem only, due to the use of an old version of uuencode that produces blank characters some times suppressed by the mailer. Same content

A Quasi-Hopf algebra interpretation of quantum 3-j and 6-j symbols and difference equations

We consider the universal solution of the Gervais-Neveu-Felder equation in the ${\cal U}_q(sl_2)$ case. We show that it has a quasi-Hopf algebra interpretation. We also recall its relation to quantum 3-j and 6-j symbols. Finally, we use this solution to build a q-deformation of the trigonometric Lam\'e equation.Comment: 9 pages, 4 figure

The bicomplex quantum Coulomb potential problem

Generalizations of the complex number system underlying the mathematical formulation of quantum mechanics have been known for some time, but the use of the commutative ring of bicomplex numbers for that purpose is relatively new. This paper provides an analytical solution of the quantum Coulomb potential problem formulated in terms of bicomplex numbers. We define the problem by introducing a bicomplex hamiltonian operator and extending the canonical commutation relations to the form [X_i,P_k] = i_1 hbar xi delta_{ik}, where xi is a bicomplex number. Following Pauli's algebraic method, we find the eigenvalues of the bicomplex hamiltonian. These eigenvalues are also obtained, along with appropriate eigenfunctions, by solving the extension of Schrodinger's time-independent differential equation. Examples of solutions are displayed. There is an orthonormal system of solutions that belongs to a bicomplex Hilbert space.Comment: Clarifications; some figures removed; version to appear in Can. J. Phy

Light-Cone Quantization of the Liouville Model

We present the quantization of the Liouville model defined in light-cone coordinates in (1,1) signature space. We take advantage of the representation of the Liouville field by the free field of the Backl\"{u}nd transformation and adapt the approch by Braaten, Curtright and Thorn. Quantum operators of the Liouville field $\partial_{+}\phi$, $\partial_{-}\phi$, $e^{g\phi}$, $e^{2g\phi}$ are constructed consistently in terms of the free field. The Liouville model field theory space is found to be restricted to the sector with field momentum $P_{+}=-P_{-}$, $P_{+}> 0$ , which is a closed subspace for the Liouville theory operator algebra.Comment: 16 p, EFI-92-6

Tariff-Rate Quotas, Rent-Shifting and the Selling of Domestic Access

Tariff-rate quotas (TRQs) have replaced quotas at the end of the Uruguay Round. We analyze TRQs when a foreign firm competes against a domestic firm in the latter’s market. Our benchmark is the strategic rent-shifting tariff. We show that the domestic price-equivalent TRQ is a better instrument welfare-wise, as it can extract all of the rents from the foreign firm. We show that different pairs of within-quota tariff and quota can support full rent extraction. The implication is that reduction of the former and enlargement of the latter, holding the above-quota tariff constant, may have no liberalizing effects. The first-best TRQ and the strategic tariff generate different prices. When firms have identical and constant marginal cost, the first-best TRQ entails selling a subsidy to the foreign firm and forcing the exit of the domestic firm.Financial Economics, International Relations/Trade, Marketing, Political Economy,

Strong-Coupling Effects in "Dirty" Superfluid 3He^3He

The contribution of the strong-coupling effects to the free energy of the "dirty" superfluid $^3He$ is estimated using a simple model. It is shown that the strong-coupling effects are less susceptible to the quasiparticle scattering events in comparison to the weak-coupling counterpart. This supports the conclusion about stabilization of the $B$-phase in aerogel environment at pressures where the $A$-phase takes over in bulk superfluid $^3He$, in accordance with recent experimental observations in zero magnetic field.Comment: 10 pages, LaTeX file. This is a revised version of the paper with some additional comments and references, and corrected typos. Submitted to Journal of Physics: Condensed Matte

Etude des interactions protéine-protéine par double hybride bactérien : Applications en agro-alimentaire et en santé

Protein-protein interaction\u27s studies are a major challenge for current research. These interactions are highly specific and regulate all processes in cell, from metabolism to responses to external stimuli. So they are crucial targets for therapeutics. Many methods are available for study them, but only few allow discovering many protein complexes together, as two-hybrid can do. The goal of this work is to underscore new partners of p21 and STAT3, two major protein involved in tumour control, and on the other side is to define the function of MtSAP1 which is involved in Medicago truncatula seed germination and in abiotic stresses\u27 response. STAT3 and p21\u27s potential interacting partners analysis strongly suggest the existence of new tumour molecular mechanisms where STAT3 plays a role, and define new hypothesis for p21\u27s function when the cell enter in a specific process. A new protein complex, p21/prohibitin2, was chosen for a detailed study but its function was only supposed: it could be a transcriptional regulator and/or could be involved in p21 proteolysis. In parallel, MtSAP1 gene study and two-hybrid analysis strongly suggest the involvement of MtSAP1 during cell response to hypoxia: MtSAP1 would be induce in response to hypoxia and then could have an important role in plant detoxification and tolerance. This work, supported by literature, underscores a link between p21, STAT3 and MtSAP1 during human tumour cell response to hypoxia. The validation of this hypothesis will let to deepen cell protection mechanisms against hypoxia, in human tumour as well as during tolerance establishment in Medicago truncatula

N=4N=4 super KdV equation

We construct $N=4$ supersymmetric KdV equation as a hamiltonian flow on the $N=4\;SU(2)$ super Virasoro algebra. The $N=4$ KdV superfield, the hamiltonian and the related Poisson structure are concisely formulated in $1D \;N=4$ harmonic superspace. The most general hamiltonian is shown to necessarily involve $SU(2)$ breaking parameters which are combined in a traceless rank 2 $SU(2)$ tensor. First nontrivial conserved charges of $N=4$ super KdV (of dimensions 2 and 4) are found to exist if and only if the $SU(2)$ breaking tensor is a bilinear of some $SU(2)$ vector with a fixed length proportional to the inverse of the central charge of $N=4\;SU(2)$ algebra. After the reduction to $N=2$ this restricted version of $N=4$ super KdV goes over to the $a=4$ integrable case of $N=2$ super KdV and so is expected to be integrable. We show that it is bi-hamiltonian like its $N=2$ prototype.Comment: 11 pages, preprint ENSLAPP-L-415-9