abstract

Kazuhiro Hikami: Hyperbolic geometry and dilogarithm

It is known that the dilogarithm function is related to the 3-dimensional hyperbolic geometry.

As an introductory lecture of the Volume Conjecture,

I will talk about the hyperbolic volume and the A-polymial of knots.

Ivan C.H. Ip: Positive Representations: Motivation, Construction and Braiding Structure

In these lectures, I will introduce the family of positive principal series representations for split real quantum groups by positive self-adjoint operators. The construction of these representations gives the starting point of a new research program devoted to the representation theory of split real quantum groups initiated in the joint work with Igor Frenkel, with a future perspective comparable to past development of finite-dimensional representation theory of quantum groups established by Drinfeld and Jimbo.

In the first lecture, I will explain the motivation in the Uq(sl(2,R)) case by the work of J. Teschner, and describe the technical ingredients involving the quantum dilogarithms and functional analysis. In the second lecture, I will construct the positive representations for arbitrary type generalizing the rank 1 theory, and explain the connection between the modular double and the Langland's dual. In the third lecture, I will explain the recent construction of the universal R operator through the language of multiplier Hopf algbera, generalizing the formula of the R matrix by Kirillov-Reshetikhin in the compact case. This construction gives the braiding structure of the positive representations.

Sergey M. Sergeev: Quantum dilogarithm and three dimensional integrable models

Most natural place of an appearance of Faddeev-Barnes quantum dilogarithm is the realm of exactly solvable models of statistical mechanics and quantum field theory in wholly discrete three dimensional space-time. I will introduce the audience to the notion of quantum dilogarithm and Kashaev's identity. Then, I will explain what is the quantum tetrahedron equation and how one can derive its solutions in therms of quantum dilogarithms. Next, I will show the relation between three dimensional models and the representation theory of modular double of quantum groups. In particular, I will explain the appearance of Faddeev-Volkov model and discuss its features. Finally, I willl compute the partition function of wholly three dimensional stat mechanical model with the weights parametrized in terms of quantum dilogarithm.

Joerg Teschner: Modular doubles of some quantum affine algebras

In my lectures I plan to discuss modular duality phenomena in the theory of quantum affine algebras: There exist infinite-dimensional representations of these algebras for which the universal R-matrix can be regularized to produce sensible R-operators acting on tensor products of such representations. As an interesting application I plan to discuss integrable lattice discretizations of the affine Toda theories.

Alexander Y. Volkov: An Introduction to Quantum Dilogarithm Identities