Structure of representations of infinite dimensional Lie algebras and conformal field theory

无限维李代数的表示结构和共形场论

• 批准号: 0500759
• 负责人: Rinat Kedem
• 金额: $ 10.59万
• 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: 2005
• 资助国家: 美国
• 起止时间: 2005-05-15 至 2008-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500759&HistoricalAwards=false
• 关键词: Structure; representations; infinite; dimensional; Lie

The main objectives of this project are explicit constructions ofrepresentations, related to the fermionic formulas for characters ofrepresentations of Virasoro or other W-algebras and affine Liealgebras. The project also uses the ideas encountered in fermionicconstructions, to give formulas for graded multiplicities ofirreducible modules in the tensor product of finite-dimensional Liealgebra modules, or integrable modules in the fusion product of affinealgebra modules (coinvariants or conformal blocks). Some of thespecific goals are: The study of fusion products of representations of finite-dimensional simple Lie algebras; theinductive limit of the fusion product as the number of factors becomesinfinite in a stabilized regime, which is expected to give newrealizations of integrable modules of affine Lie algebras;combinatorial identities for q-series which result from theseconstructions; semi-infinite constructions of arbitrary highestweight representations of affine Lie algebras modules; andapplications of semi-infinite (c.f. Feigin and Styoanovskii's approach)constructions in solutions of the fractional quantum Hall effect.This research is at the interface of combinatorial representationtheory and mathematical physics. The constructions are guided byconformal field theory and exactly solvable models in statisticalmechanics. The results are important in the representation theory ofLie algebras and affine Lie algebras. Combinatorial questions such asdimensions of weight spaces in irreducible representations,multiplicities of irreducible components in tensor products ofLie-algebra modules are related to the counting of certain matrixelements or conformal blocks. The physical applications include thestudy of wave functions in the quantum Hall effect, and partition functions in statistical mechanical systems at criticality.

这个项目的主要目标是显式构造表示，涉及Virasoro或其他W-代数和仿射李代数表示特征的费米子公式。该项目还使用费米子构造中遇到的思想，给出有限维李代数模的张量积中不可约模的分次多重性公式，或仿射代数模（共不变量或共形块）融合积中可积模的公式。其中一些具体目标是：有限维单李代数表示的融合积的研究，在稳定化区域中因子个数为无穷大时融合积的归纳极限，期望给出仿射李代数可积模的新的实现，由这些构造得到的q-级数的组合恒等式，仿射李代数模的任意最高权表示的半无限构造，仿射李代数模的任意最高权表示的半无限以及半无限（c.f. Feigin和Styoanovskii的方法）的分数量子霍尔效应的解决方案的建设。这项研究是在组合表示理论和数学物理的接口。构造的指导思想是共形场论和弹性力学中的精确可解模型。这些结果在李代数和仿射李代数的表示理论中具有重要意义。不可约表示中权空间的维数、李代数模张量积中不可约分量的重数等组合问题都与某些矩阵元素或共形块的计数有关。其物理应用包括量子霍尔效应中的波函数研究，以及临界统计力学系统中的配分函数研究。