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

这个项目的主要目标是显式构造表示，与Virasoro或其他W-代数和仿射Lie代数的表示的特征的费米子公式有关。该项目还利用费米构造中的思想，给出了有限维Lie代数模的张量积中的可约模或仿射代数模(余不变量或保形块)的融合积中的可积模的分次重数的公式。一些具体的目标是：研究有限维单李代数表示的融合乘积；融合乘积的归纳极限当因子的个数在稳定的区域内变得无限时，有望给出仿射李代数的可积模的新的实现；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；以及半无限(c.f.Feigin和Styoanovskii的方法)构造分数量子霍尔效应的解。这项研究处于组合表示理论和数学物理的交界处。这种构造是由保形场理论和统计力学中的精确可解模型指导的。这些结果在Lie代数和仿射李代数的表示理论中具有重要意义。Lie-代数模的不可约表示中权空间的维数、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。物理应用包括研究量子霍尔效应中的波函数，以及临界点统计力学系统中的配分函数。

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.