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We prove a full global Jacquet-Langlands correspondence between GL(n) and division algebras over global fields of non zero characteristic. If $D$ is a central division algebra of dimension $n^2$ over a global field $F$ of non zero characteristic, we prove that there exists an injective map from the set of automorphic square integrable representations of the multiplicative group of $D$ to the set of automorphic square integrable representations of GL_n(F), compatible at all places with the local Jacquet-Langlands correspondence for unitary representations. We characterize the image of the map. As a consequence we get multiplicity one and strong multiplicity one theorems for the multiplicative group of D., Comment: 29 pages. We have added references, corrected misprints and corrected a proof

For a finite dimensional simple Lie algebra g, the standard universal solution R(x) in $U_q(g)^{\otimes 2}$ of the Quantum Dynamical Yang--Baxter Equation can be built from the standard R--matrix and from the solution F(x) in $U_q(g)^{\otimes 2}$ of the Quantum Dynamical coCycle Equation as $R(x)=F^{-1}_{21}(x) R F_{12}(x).$ It has been conjectured that, in the case where g=sl(n+1) n greater than 1 only, there could exist an element M(x) in $U_q(sl(n+1))$ such that $F(x)=\Delta(M(x)){J} M_2(x)^{-1}(M_1(xq^{h_2}))^{-1},$ in which $J\in U_q(sl(n+1))^{\otimes 2}$ is the universal cocycle associated to the Cremmer--Gervais's solution. The aim of this article is to prove this conjecture and to study the properties of the solutions of the Quantum Dynamical coBoundary Equation. In particular, by introducing new basic algebraic objects which are the building blocks of the Gauss decomposition of M(x), we construct M(x) in $U_q(sl(n+1))$ as an explicit infinite product which converges in every finite dimensional representation. We emphasize the relations between these basic objects and some Non Standard Loop algebras and exhibit relations with the dynamical quantum Weyl group., Comment: 46 pages

We study the quantization of Chern-Simons theory with group $G$ coupled to dynamical sources. We first study the dynamics of Chern-Simons sources in the Hamiltonian framework. The gauge group of this system is reduced to the Cartan subgroup of $G.$ We show that the Dirac bracket between the basic dynamical variables can be expressed in term of dynamical $r-$matrix of rational type. We then couple minimally these sources to Chern-Simons theory with the use of a regularisation at the location of the sources. In this case, the gauge symmetries of this theory split in two classes, the bulk gauge transformation associated to the group $G$ and world lines gauge transformations associated to the Cartan subgroup of $G$. We give a complete hamiltonian analysis of this system and analyze in detail the Poisson algebras of functions invariant under the action of bulk gauge transformations. This algebra is larger than the algebra of Dirac observables because it contains in particular functions which are not invariant under reparametrization of the world line of the sources. We show that the elements of this Poisson algebra have Poisson brackets expressed in term of dynamical $r-$matrix of trigonometric type. This algebra is a dynamical generalization of Fock-Rosly structure. We analyze the quantization of these structures and describe different star structures on these algebras, with a special care to the case where $G=SL(2,{\mathbb R})$ and $G=SL(2,{\mathbb C})_{\mathbb R},$ having in mind to apply these results to the study of the quantization of massive spinning point particles coupled to gravity with a cosmological constant in 2+1 dimensions., Comment: 32 pages and 1 eps figure

We study the Hamiltonian formulation of Plebanski theory in both the Euclidean and Lorentzian cases. A careful analysis of the constraints shows that the system is non regular, i.e. the rank of the Dirac matrix is non-constant on the non-reduced phase space. We identify the gravitational and topological sectors which are regular sub-spaces of the non-reduced phase space. The theory can be restricted to the regular subspace which contains the gravitational sector. We explicitly identify first and second class constraints in this case. We compute the determinant of the Dirac matrix and the natural measure for the path integral of the Plebanski theory (restricted to the gravitational sector). This measure is the analogue of the Leutwyler-Fradkin-Vilkovisky measure of quantum gravity., Comment: 25 pages, no figures, references added

We give a short summary of results and conjectures in the theory of dynamical quantum group related to the dynamical coboundary equation also known as IRF-Vertex transform. O.Babelon has shown that the dynamical twist $F(x)$ of $U_q(sl(2))$ is a dynamical coboundary $M(x)$ i.e $F(x)M_1(xq^{h_2})M_2(x)=\Delta(M(x)).$ We give a new formula for this element $M(x)$ as an infinite product and give a new proof of the coboundary relation. Our proof involves the quantum Weyl group element, giving possible hint for the generalization to higher rank case., Comment: 9 pages in latex. Proceedings of the conference ``Recent Advances in the Theory of Quantum Integrable Systems 2003'', LAPTH, Annecy-Le-Vieux remarks added to clarify the main proof

We give the complete set of irreducible representations of U(SU(2))_q when q is a m-th root of unity. In particular we show that their dimensions are less or equal to m. Some of them are not highest weight representations., Comment: 9 pages, plain TeX ; due to the renewed interest in quantum groups at roots of unity, we put this paper in the archive

We study the tensor product of principal unitary representations of the quantum Lorentz group, prove a decomposition theorem and compute the associated intertwiners. We show that these intertwiners can be expressed in terms of complex continuations of 6j symbols of U_q(su(2)). These intertwiners are expressed in terms of q-Racah polynomials and Askey-Wilson polynomials. The orthogonality of these intertwiners imply some relation mixing these two families of polynomials. The simplest of these relations is the orthogonality of Askey-Wilson polynomials., Comment: 34 pages LaTex file, no figures

We construct a universal trigonometric solution of the Gervais-Neveu-Felder equation in the case of finite dimensional simple Lie algebras and finite dimensional contragredient simple Lie superalgebras., Comment: 12 pages, LaTeX2e with packages vmargin, wasysym, amsmath, amssymb

This work begins with a review of complexification and realification of Hopf algebras. We emphasize the notion of multiplier Hopf algebras for the description of different classes of functions (compact supported, bounded, unbounded) on complex quantum groups and the construction of the associated left and right Haar measure. Using a continuation of $6j$ symbols of $SU_q (2)$ with complex spins, we give a new description of the unitary representations of $SL_q (2,\CC)_{\RR}$ and find explicit expressions for the characters of $SL_q (2,\CC)_{\RR}$. The major theorem of this article is the Plancherel theorem for the Quantum Lorentz Group., Comment: 60 pages, tared gzipped Postscript file, major revision of the previous version, the Plancherel theorem is established in the more general sense and we delay the study of Fusion theory to the next part of this paper

We define and study the properties of observables associated to any link in $\Sigma\times {\bf R}$ (where $\Sigma$ is a compact surface) using the combinatorial quantization of hamiltonian Chern-Simons theory. These observables are traces of holonomies in a non commutative Yang-Mills theory where the gauge symmetry is ensured by a quantum group. We show that these observables are link invariants taking values in a non commutative algebra, the so called Moduli Algebra. When $\Sigma=S^2$ these link invariants are pure numbers and are equal to Reshetikhin-Turaev link invariants., Comment: 39, latex, 7 figures