The Jacobson-Morozov theorem for Lie superalgebras via semisimplification functor for tensor categories

Abstract: The celebrated Jacobson-Morozov theorem claims that every nilpotent element of a semisimple Lie algebra g can be embedded into an sl(2)-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. We also discuss possible generalization of reductive envelope of an algebraic group to the case of a supergroup. (Joint work with Inna Entova-Aizenbud).

Mathematics

Audience: researchers in the topic

T-Rep: A midsummer night's session on representation theory and tensor categories

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