Contragredient Lie algebras in symmetric categories

Abstract

We define contragredient Lie algebras in symmetric categories, generalizing the construction of Lie algebras of the form $\mathfrak{g}(A)$ for a Cartan matrix $A$ from the category of vector spaces to an arbitrary symmetric tensor category. The main complication resides in the fact that, in contrast to the classical case, a general symmetric tensor category can admit tori (playing the role of Cartan subalgebras) which are non-abelian and have a sophisticated representation theory. Using this construction, we obtain and describe new examples of Lie algebras in the universal Verlinde category in characteristic $p\ge 5$. We also show that some previously known examples can be obtained with our construction.

42 pages. v2: new version of Lemma 5.9. We added Example 5.12