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

在此次助学金下，我与Victor G.Kac教授共同进行了研究，并取得了以下成果：仿射超代数的“可积表示”从来不是一个简单的概念，但应该仔细对待。我们发现可积表示由两类表示组成，即主可积表示和次主可积表示，并给出了主可积模块和次主可积模块的所有最高权值的显式完整列表。利用自由玻色子和自由费米子给出了基本模sl(m|n)^-和osp(m|n)^-的显式构造。利用这种显式结构，我们计算了特征，得到了三种特征公式——Weyl-Kac型、theta-function型和准粒子型。从这些特征公式中，我们发现基本的sl(m|1)^-模的特征是阿佩尔的椭圆函数，这是阿佩尔在19世纪80年代发现的，但已经被遗忘了一百多年。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)^的平凡表示是临界能级的表示，因为它的对偶Coxeter等于0。所以这些超代数没有已知的分母恒等式。利用黎曼关系，我们显式地得到了sl(2 bb0 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不能给出正确的w代数，而与仿射超代数相关的w代数的构造一直是一个问题。我们成功地解决了这个困难，通过张紧因子，这是由代数变化引起的，用通常的brst复合体。用这种方法得到的仿射超代数sl(2|1)^的w代数是无中心Virasoro代数与N=2超共形代数的直接和。这个理论使我们能够用sl(2|1)^的可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，还存在N=2个表示的奇特级数，它们的特征是半模函数。这项研究正在深入进行中。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