Quantum Groups, Integrable Models and Statistical Systems

Fedor Aleksandrovich Smirnov

Leon A. Takhtajan

Jean LeTourneux, Luc Vinet

This volume contains the lectures presented at the workshop entitled Quantum Groups, Integrable Models and Statistical Systems. The papers give either a full exposition of original results or a review of fundamental aspects of this active research area.

Anjan Kundu

Basic concepts of quantum integrable systems (QIS) are presented stressing on the unifying structures underlying such diverse models. Variety of ultralocal and nonultralocal models is shown to be described by a few basic relations defining novel algebraic entries. Such properties can generate and classify integrable models systematically and also help to solve exactly their eigenvalue problem in an almost model-independent way. The unifying thread stretches also beyond the QIS to establish its deep connections with statistical models, conformal field theory etc. as well as with abstract mathematical objects like quantum group, braided or quadratic algebra

Revista, Zen, MATH, 10

This paper rigorously investigates the profound interplay between quantum group symmetries and the deformed spectral transform, particularly in the context of soliton theory. We delve into how the introduction of a deformation parameter, characteristic of quantum groups, modifies the fundamental algebraic structures underlying integrable systems and, consequently, the spectral properties of their soliton solutions. By employing the framework of the quantum inverse scattering method (QISM), we construct deformed Lax pairs and analyze their impact on the scattering data. Our analysis reveals that quantum group symmetries induce a non-trivial deformation of the scattering matrix and its associated Jost functions, leading to quantized spectral parameters and modified conservation laws. We demonstrate how this deformation manifests in the properties of solitons, suggesting a departure from classical integrability and opening avenues for understanding quantum integrable models. The implications extend to a deeper understanding of quantum field theories and statistical mechanics models where these symmetries are paramount.