Affine algebra representations and discrete integrability

仿射代数表示和离散可积性

• 批准号: 1100929
• 负责人: Rinat Kedem
• 金额: $ 14.43万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1100929&HistoricalAwards=false
• 关键词: Affine; algebra; representations; discrete; integrability

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.

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论，以及（3）动态系统的组合数学（例如簇代数）之间的关系。从物理角度来看，该项目可应用于弦理论中的穿壁公式以及凝聚态物理中负责量子霍尔效应的准粒子的波函数。研究的问题包括仿射李代数模及其融合积的费米子构造；应用统计物理学方法对与离散可积系统（例如T系统或Q系统）相关的簇代数中的簇变量给出显式解，从而证明相关的正性猜想；以及这些系统的非交换概括，与 Kontsevich 非交换穿墙公式或描述量子离散 Liouville 或 Hirota 方程的量子簇代数相关。这项研究处于数学和物理之间的边界。它寻求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问题。这些问题的主要特征是可积性，即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分区函数的形式找到显式解，这些解显然是系统配置的正和。这种正性属性通常是基础数学对象的推测属性。