Universal tensor categories, representations of infinite-dimensional Lie algebras, and infinite-dimensional geometry

通用张量范畴、无限维李代数的表示和无限维几何

• 批准号: 518961449
• 负责人: Professor Dr. Ivan Penkov
• 金额: --
• 依托单位: Department of Mathematics and Logistics
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/518961449?language=en
• 关键词: Universal; tensor; categories; representations; infinite

This is a broad proposal in the general field of infinite-dimensional Lie representation theory and the related infinite-dimensional geometry. It builds upon recent advances in the author’s program in this field going back for up to twenty years. The concrete topics of study are spread quite equally between algebraic and geometric representation theory. On the algebraic side, we plan to investigate the structure and representation theory of infinite-dimensional Mackey Lie algebras. We also propose to study an interesting and quite intricate category of sl(∞)-modules. Next, our proposed investigation of the isomorphism classes of automorphism groups of ind-varieties of generalized flags has its origin in geometry but is in fact a mostly algebraic study. In addition, we intend to pursue two geometric directions of research. One is the study of homogeneous ind-spaces of diagonal ind-groups, with applications of ind-varieties of generalized flags. The other is the introduction and subsequent study of thick flag varieties for the ind-group GL(∞). This latter theory is motivated by Kashiwara’s theory of thick flag varieties for a Kac- Moody Lie algebra.

这是在无限维李氏表示理论和相关无限维几何的一般领域中的一个广泛的建议。它建立在作者的程序在这个领域的最新进展可以追溯到20年。具体的研究课题在代数表示理论和几何表示理论之间分布得相当均匀。在代数方面，我们计划研究无限维麦基李代数的结构和表示理论。我们也提出研究一个有趣且相当复杂的sl(∞)-模的范畴。其次，我们提出的关于广义标志的非变异自同构群的同构类的研究有其几何起源，但实际上主要是一个代数研究。此外，我们打算追求两个几何方向的研究。一是研究对角旋群的齐次旋群空间，以及广义旋群的旋群变异的应用。二是对GL（∞）群的厚标志变体的介绍和后续研究。后一种理论是由Kashiwara的Kac- Moody李代数的厚旗变体理论所推动的。