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

这个项目的主要目标是表示的显式构造，与Virasoro或其他w -代数和仿射李代数的表示特征的费米公式有关。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块张量积中不可约模块的梯度多重性公式，或仿射代数模块（协不变或共形块）融合积中的可积模块的梯度多重性公式。一些具体的目标是：研究有限维简单李代数表示的融合积；在稳定状态下，当因子数变为无穷大时，融合积的归纳极限，有望给出仿射李代数的可积模的新实现；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半无限结构（c.f Feigin和Styoanovskii的方法）在分数量子霍尔效应解中的应用。本研究是组合表示理论和数学物理的交叉领域。这些构造以统计力学中的共形场论和精确可解模型为指导。所得结果在李代数和仿射李代数的表示理论中具有重要意义。组合问题，如不可约表示中权重空间的维数，李代数模块张量积中不可约分量的复数，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和临界状态下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.