Lie algebras in symmetric tensor categories

Abstract

We develop the general theory of Lie algebras in symmetric tensor categories. We prove categorical versions of classical results such as Engel's theorem, Cartan's criterion, and that all derivations of a strongly semisimple Lie algebra are inner. For finite-dimensional graded Lie algebras, we construct a graded category $\mathcal{O}$, classify its graded-simple objects, and prove the generic graded irreducibility of Verma modules. Lastly, we develop the theory of root systems for contragredient Lie algebras.

47 pages