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

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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.