Groupes quantiques associes aux courbes rationnelles et elliptiques et leurs applications

TL;DR

This thesis explores the structure and applications of elliptic quantum groups related to rational and elliptic curves, focusing on their connections to integrable systems, statistical models, and the construction of partition functions.

Contribution

It provides a comparative analysis of elliptic quantum groups, interprets the Toda chain transition function via Lax operators, and constructs SOS model partition functions using elliptic quantum groups.

Findings

Elliptic quantum groups belong to different bialgebra categories.

The transition function of the periodic Toda chain can be interpreted through rational Lax operators.

Partition functions of the SOS model can be constructed from elliptic quantum group projections.

Abstract

The thesis was defended by the author in University of Angers (France). It consists of four parts. The fist part (in French) is introductory and is devoted to relation between quantum groups, integrable systems and statistical models. In the second part (in English) the transition function of the periodic Toda chain is interpreted in terms of the formalism of rational Lax operators. In the third part (in French) one compares two elliptic quantum groups and one conclude that they belong to two different bialgebra categories. The fourth part (in English) contains a construction of the partition function of the SOS model in terms of the projections of an elliptic quantum group.