Structural and geometric study of representations with applications to categorification

表示的结构和几何研究及其在分类中的应用

• 批准号: 124681545
• 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: Priority Programmes
• 财政年份: 2009
• 资助国家: 德国
• 起止时间: 2008-12-31 至 2016-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/124681545?language=en
• 关键词: Structural; geometric; study; representations; applications

This project is the continuation of the project “Structure and representations of Infinite- and finite-dimensional Lie algebras" with principal investigator I. Penkov funded in the first phase of the project “Darstellungstheorie". In the last three years the goals of this latter project have been fully achieved, in particular essential progress in several directions of study has been made by I. Penkov and his collaborators, see section 2.1 below. The continuation of the project is enriched by the addition of a joint supervision doctoral project of I. Penkov and A. Huckleberry on the geometry of open orbits of real forms of semisimple Lie groups on ag manifolds and the representation theory related to the cycle spaces of these orbits. This project is to be carried out by a doctoral student in 2012 through 2015. In addition, the collaboration of I. Penkov with Vera Serganova (UC Berkeley) and Gregg Zuckerman (Yale University) should continue and build upon results obtained during the first phase of funding. Concretely, this involves the study of Koszulity of categories of integrable representations of infinite-dimensional Lie algebras and Lie superalgebras, work on a conjecture that the Zuckerman functor establishes an equivalence between certain categories of lowest weight modules and certain categories of generalized Harish-Chandra modules. Furthermore I. Penkov, V. Serganova and Igor Frenkel (Yale University) plan to work also on a categorification of the boson-fermion correspondence. Finally, a joint seminar, with the group of Peter Heinzner in Bochum, on the interaction of complex geometry and algebraic geometry in the context of geometric representation theory is planned.

本项目是与首席研究员I。Penkov资助了“Darstellungstheorie”项目的第一阶段。在过去的三年里，后一个项目的目标已经完全实现，特别是我在几个研究方向上取得了重要进展。Penkov及其合作者，见下文第2.1节。该项目的继续是丰富的，增加了一个联合监督博士项目的我。Penkov和A. Huckleberry关于ag流形上半单李群的真实的形式的开轨道的几何学以及与这些轨道的循环空间有关的表示论。该项目将于2012年至2015年由一名博士生实施。此外，I. Penkov与Vera Serganova（加州大学伯克利分校）和Gregg Zuckerman（耶鲁大学）应该继续并建立在第一阶段资助期间取得的成果基础上。具体地说，这涉及到无限维李代数和李超代数的可积表示范畴的Koszulity的研究，工作在一个猜想，即Zuckerman函子建立了某些类别的最低权模和某些类别的广义Harish-Chandra模之间的等价性。此外，I. Penkov，V. Serganova和Igor Frenkel（耶鲁大学）也计划对玻色子-费米子对应进行分类。最后，计划与波鸿的Peter Heinzner小组联合举办一次讨论会，讨论在几何表示理论的范围内复几何和代数几何的相互作用。