Affine algebra representations and discrete integrability

该研究项目利用（1）统计力学中的离散可积模型，（2）仿射和量子仿射代数的表示理论，以及（3）动力系统（如簇代数）的组合学之间的关系。从物理学的角度来看，该项目已应用于弦理论中的跨壁公式和凝聚态物理学中负责量子霍尔效应的准粒子波函数。研究的问题包括：仿射李代数模及其融合积的费米子构造;应用统计物理的方法给出与离散可积系统（如T系统或Q系统）相关的簇代数中簇变量的显式解，从而证明相关的正性定理;以及这些系统的非对易推广，与Kontsevich非对易跨壁公式有关，或者描述量子离散Liouville或Hirota方程的量子簇代数，这一研究处于数学和物理的边界。它旨在应用统计力学的技术来解决仿射李代数及其量化的组合学和表示论中的问题。这些问题的主要特征是可积性，也就是说，它们产生于具有高度对称性的系统。通常这种对称性允许以物理配分函数的形式找到显式解，这些物理配分函数在系统的配置上是明显的正和。这种积极性的性质常常是数学对象的一种固有性质。

The research project exploits relations among (1) discrete integrable models in statistical mechanics, (2) representation theory of affine and quantum affine algebras, and (3) the combinatorics of dynamical systems such as cluster algebras. From the physical point of view, the project has applications to wall crossing formulas in string theory and the wave functions of quasi-particles responsible for the quantum Hall effect in condensed matter physics. The problems investigated include fermionic constructions of affine Lie algebra modules and their fusion products; Applications of methods from statistical physics to give explicit solutions for the cluster variables in cluster algebras related to discrete integrable systems such as T-systems or Q-systems, thereby giving proofs of the relevant positivity conjectures; and non-commutative generalizations of these systems, related to the Kontsevich non-commutative wall-crossing formula, or the quantum cluster algebras which describe quantum discrete Liouville or Hirota equations.This research is at the boundary between mathematics and physics. It seeks to apply techniques from statistical mechanics to solving problems in combinatorics and representation theory of affine Lie algebras and their quantization. The prime characteristic of these problems is integrability, that is, they arise from systems with a high degree of symmetry. Often this symmetry allows for finding explicit solutions in the form of physical partition functions, which are manifestly positive sums over configurations of a system. This positivity property is frequently a conjectured property of the underlying mathematical objects.