Conformal Field Theories, Links and Quantum Groups

In chapter 1 we recall briefly some aspects of Hopf algebras, quantum groups and their representations. In chapter 2 we describe some basic aspects of conformal field theory, with particular emphasis on the presentation of rational conformal field theories given by Moore and Seiberg which is, in our opinion, the closest one to the notion of quasi-Hopf algebras. In chapter 3 we describe link-diagrams on any 2-dimensional surface I: and we study their properties. More precisely we consider the module over some polynomial ring generated by link-diagrams and seek the conditions under which this module can be given the structure of a coalgebra or of an algebra. In chapter 4 we use the algebraic structures considered in chapter 3 in order to construct link-invariants for links in any manifold of the type I: x [O, 1]. This already gives an extension of the Jones polynomial when I: is not the disc. But we can go one step further. We extend also the Homfly polynomial (when I: is an open Riemann surface) obtaining a four variables link-polynomial. We also show why link-invariants are related to the Yang-Baxter equations. In this respect we introduce the concept of quantum-holonomy for a link, which is a special kind of partition function. We discuss the condition under which the quantum holonomy reproduces exactly some of the characteristics of the Witten's functional integral. Also we consider the quantum-holonomy related to the Drinfeld quasi Hopf algebra and show how to obtain a quantization of the Goldman Lie-algebra of loops on a surface. Finally m chapter 5 we construct using the Faddeev-Reshetikhin-Taktajan method [30] the quantum group, corresponding to the Yang-Baxter matrix related to link-diagrams. This quantum group is a multiparameter deformation of Uq(sl(n) (see also [31] ). There are many interesting differences between the multiparameter quantum-group and the ordinary (one-parameter) quantum group. These differences include the existence of a non-central quantum determinant and the doubling of the number of the generator of the quantum Cartan subalgebra. In chapter 6 we discuss in detail Drinfeld's quasi-Hopf algebras and the two representations of the braid group which are shown to be equivalent by Drinfeld-Kohno theorem. The relation between rational conformal field theories and quasi-Hopf algebras is finally discussed.