Representation theory of infinite-dimensional Lie algebras and superalgebras and its mathematical applications

在此助学金下，我与维克托·G·卡克教授进行了联合研究，取得了以下成果：1.仿射超代数的“可积表示”从来都不是一个容易的概念，但应该谨慎对待。我们发现可积表示由两类组成，即主次可积表示和次主可积表示，并给出了主次主可积模的所有最高权的显式和完备表。我们利用自由玻色子和自由费米子给出了基本sl(m|n)^-模和osp(m|n)^-模的显式构造。利用这种显式结构，我们计算了特征标，得到了三种特征标公式-Weyl-Kac型、theta函数型和准粒子型。从这些特征标公式中我们发现，基本sl(m|1)^-模的特征标是由Appell在1880年的S发现的，但已被…遗忘了一百多年更多。这些函数不是模函数，但我们成功地计算了它们的渐近性。仿射超代数sl(2|2)^的平凡表示是临界值的表示，因为它的对偶Coxeter等于0。因此，这种超代数没有已知的分母恒等式。我们利用Riemann的theta关系，显式地得到了sl(2|2)^的分母公式。根据Frenkel-Kac-Wakimoto(1994)的理论，通常仿射李代数的W-代数及其表示都是根据量子化的Drinfeld-Sokolov约化构造的。但是，对于仿射超代数，这种方法的直接推广并不能给出一个正确的W-代数，而与仿射超代数相关的W-代数的构造长期以来一直是一个问题。我们成功地解决了这一困难，将由代数簇产生的因子与通常的BRST-复形张开。用这种方法得到的仿射超代数sl(2|1)^的W-代数是无心Virasoro代数和N=2超共形代数的直和。这一理论使我们能够通过sl(2|1)^的容许表示来详细研究N=2超共形代数的表示。实际上，我们发现，除了通常的最小级数表示外，还有一系列奇怪的N=2表示，它们的特征是半模函数。这项研究目前正在非常密集地进行中。较少

Under this Grant-in-Aid, I made joint research with Professor Victor G.Kac, and obtained the following results :1. "Integrable representations" for affine superalgebras are never easy concept, but should be treated carefully. We found that integrable representations consist of two kinds, namely principal-integrable representations and subprincipal-integrable representations, and gave the explicit and complete list of all highest weights for principal and subprincipal integrable modules.2. We gave an explicit construction of fundamental sl(m|n)^- and osp(m|n)^-modules by using free bosons and free fermions. Using this explicit construction, we calculated the characters and obtained three kinds of character formulas --- Weyl-Kac type, theta-function type and quasi-particle type. From these character formulas, we found that the characters of fundamental sl(m|1)^-modules are Appell's elliptic functions which were discovered by Appell in 1880's but have been forgotten over one hundred years … More . These functions are not modular functions, but we succeeded to compute their asymptotics.3. The trivial representation of an affine superalgebra sl(2|2)^ is a representation of critical level, since its dual Coxeter is equal to 0. So there was no known denominator identity for such superalgebras. We obtained explicitly the denominator formula for sl(2|2)^ by using Riemann's theta relations.4. It is known by the theory of Frenkel-Kac-Wakimoto (1994) that the W-algebra of an usual affine Lie algebra and its representations are constructed in terms of the quantized Drinfeld-Sokolov reduction. But, for affine superalgebras, an immediate extension of this method fails to give a right W-algebra, and the construction of the W-algebra associated to affine superalgebras has long been a problem. We succeeded to resolve the difficulty by tensoring the factor, which arises from the algebraic variety, with the usual BRST-complex. The W-algebra of an affine superalgebra sl (2|1)^ obtained by this method is the direct sum of the centerless Virasoro algebra and the N=2 superconformal algebra. This theory enables us to make a detail investigation on representations of the N=2 superconformal algebra by means of admissible representations of sl(2|1)^. Actually we found that, other than the usual minimal series representations, there exist curious series of N=2 representations whose characters are half-modular functions. This research is now in progress very intensively. Less