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NSF-BSF: Categorical Methods in Representation Theory of Lie Superalgebras

NSF-BSF：李超代数表示论中的分类方法

基本信息

批准号：
2001191
负责人：
Vera Serganova
金额：
$ 29.46万
依托单位：
University of California-Berkeley
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001191&HistoricalAwards=false
关键词：
NSF BSF Categorical Methods Representation

项目摘要

This project will explore connections between representation theory of Lie superalgebras and tensor categories. Representation theory of groups and Lie algebras is a mathematical language for studying symmetries of different problems in natural sciences. In this project, the PI will study a certain modification of Lie algebras called Lie superalgebras. The term "super" means that in addition to the usual (even or "bosonic") symmetries we allow another kind: "odd symmetries", which behave in a certain sense "like fermions." Superalgebras are mathematical tools encoding the concept of supersymmetry developed in physics in the second half of the 20th century. The theory of symmetric monoidal (tensor) categories is an advanced framework in representation theory. Recent developments have uncovered deep connections between superalgebras and tensor categories. The PI will study applications of these results that will shed new light on the theory of Lie superalgebras, and provide new tools and techniques for investigating supersymmetries. This research area has many open questions that are accessible accessible for beginning researchers and this proposal includes research projects to be undertaken by undergraduate and graduate students.This proposal was submitted in response to the Dear Colleague Letter NSF 17-120: Special Guidelines for Submitting Collaborative Proposals under National Science Foundation (NSF) and US-Israel Binational Science Foundation (BSF) Collaborative Research Opportunities, and the proposed research will be pursued in collaboration with an Israeli researcher. Three projects are proposed. In the first project, the PIs will describe the conditions under which a natural generalization of the Jacobson-Morozov lemma holds for supergroups (and Lie superalgebras). This will be done using the theory of semisimplifications of tensor categories, in the spirit of recent work of Etingof, Ostrik, Andre, and Kahn. The PIs will then investigate the case when the condition on the supergroup does not hold and develop an analogue of the notion of a reductive envelope for supergroups. The second project concerns symmetric monoidal functors between representations of different Lie superalgebras. The PIs will study compositions of such functors and endomorphisms of such compositions. The third project concerns the periplectic supergroup, which is an interesting superanalogue of the orthogonal group. The PIs will answer open questions about representations of the periplectic supergroups, such as computing dimensions and characters of irreducible representations, as well as use Duflo-Serganova functors to study stabilization patterns in periplectic representation theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

这个项目将探索李超代数的表示理论和张量范畴之间的联系。群和李代数的表示论是研究自然科学中不同问题对称性的一种数学语言。在这个项目中，PI将研究李代数的一种修改，称为李超代数。术语“超级”意味着，除了通常的(偶数或“玻色子”)对称性之外，我们还允许另一种对称性：“奇特对称性”，它在某种意义上表现为“费米子”。超代数是20世纪下半叶物理学中发展起来的编码超对称概念的数学工具。对称么半(张量)范畴理论是表示论中的一个高级框架。最近的发展揭示了超代数和张量范畴之间的深层次联系。PI将研究这些结果的应用，这将为李超代数理论提供新的启示，并为研究超对称提供新的工具和技术。这一研究领域有许多开放的问题，初学者可以接触到，这项建议包括本科生和研究生进行的研究项目。这项建议是为了回应尊敬的同事信NSF 17-120：在国家科学基金会(NSF)和美以双国科学基金会(BSF)合作研究机会下提交合作建议的特别指导方针，建议的研究将与一名以色列研究人员合作进行。提出了三个项目。在第一个项目中，PI将描述对超群(和李超代数)来说，Jacobson-Morosov引理的自然推广成立的条件。这将使用张量范畴的半简化理论来完成，本着Etingof，Ostrik，Andre和Kahn最近的工作精神。当超群的条件不成立时，PI将调查这种情况，并开发出类似于超群的约化包络的概念。第二个项目涉及不同李超代数的表示之间的对称么整函子。PI将研究这些函子的组成和这些组成的自同态。第三个项目涉及到环绕超群，它是正交群的一个有趣的超模拟。PI将回答有关盘周超群表示的公开问题，例如计算不可约表示的维度和特征，以及使用Duflo-Serganova函子来研究盘周表示理论中的稳定模式。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。