Quantum Groups and Functional Relations: A synthesis of methods for integrable systems

我们计划加强和扩展我们在量子可积和与量子环代数相关的可积随机系统的数学结构方面的工作。我们将开发解决这类物理系统的光谱问题的方法，并精确计算它们的热力学量和相关函数。要使用和进一步发展的主要工具将是函数方程系统，如astq -方程、Q-和y -系统、反转关系和用于计算形式因子和相关函数的各种类型的量子Knizhnik-Zamolodchikov方程。我们将特别注意区分基于代数的普遍性质和基于表示法的物理性质。前者对于函数方程的推导是充分的，后者对于求具体解是重要的。重点将放在现有的泛函方程推广到高阶代数和与系列量子环代数相关的系统，而不是a型（B-， C-和d -系列）。在应用方面，我们计划探讨泛函方程在研究可积随机系统中的适用性。

We plan to intensify and extend our work on the mathematical structure ofquantum integrable and integrable stochastic systems associated with quantumloop algebras. We will develop methods for solving the spectral problem forthese classes of physical systems and for the exact calculation of theirthermodynamic quantities and correlation functions. The central tool to beused and further developed will be systems of functional equations such asTQ-equations, Q- and Y-systems, inversion relations and various types ofquantum Knizhnik-Zamolodchikov equations for the calculation of form factorsand correlation functions. We shall pay special attention to distinguishuniversal properties, based on the algebra, from physical properties, based onthe representation. The former are supposed to be sufficient for thederivation of the functional equations, the latter become important for thespecific solutions. Emphasis will be put on a generalization of existingfunctional equations to higher rank algebras and to systems associated withquantum loop algebras of series other than the A-type (the B-, C- andD-series). On the side of applications we plan to explore the applicability offunctional equations in the study of integrable stochastic systems.