The famous Jacobson-Morozov theorem claims that every nilpotent element of a semisimple Lie algebra 𝔤 can be embedded into an 𝔰𝔩2-triple inside 𝔤. Let 𝔤 be a Lie superalgebra with reductive even part and x be an odd element of 𝔤 with non-zero nilpotent [x,x]. We give necessary and sufficient condition when x can be embedded in 𝔬𝔰𝔭(1|2) inside 𝔤. The proof follows the approach of Etingof and Ostrik and involves semisimplification functor for tensor categories. Next, we will show that for every odd x in 𝔤 we can construct a symmetric monoidal functor between categories of representations of certain superalgebras. We discuss some properties of these functors and applications of them to representation theory of superalgebras with reductive even part. (Joint work with Inna Entova-Aizenbud).
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
