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.

这是无限维李表示理论和相关的无限维几何的一般领域中的一个广泛的建议。它建立在作者的计划在这一领域的最新进展可追溯到二十年。具体的研究课题是传播相当平等之间的代数和几何表示理论。在代数方面，我们计划研究无限维Mackey李代数的结构和表示理论。我们还打算研究sl（∞）-模的一个有趣而复杂的范畴。其次，我们提出的调查的同构类的自同构群的ind-varieties的广义标志有其起源的几何，但实际上是一个主要的代数研究。此外，我们打算追求两个几何方向的研究。一个是对角ind-群的齐性ind-空间的研究，以及广义标志的ind-簇的应用。二是GL（∞）型厚旗品种的引种和后续研究。这后一种理论的动机是由Kashiwara的理论厚旗品种的卡茨-穆迪李代数。