New Constructions of Exceptional Simple Lie Superalgebras with Integer Cartan Matrix in Characteristics 3 and 5 via Tensor Categories

Abstract

AbstractUsing tensor categories, we present new constructions of several of the exceptional simple Lie superalgebras with integer Cartan matrix in characteristicp= 3 andp= 5 from the complete classification of modular Lie superalgebras with indecomposable Cartan matrix and their simple subquotients over algebraically closed fields by Bouarroudj, Grozman, and Leites in 2009. Specifically, letαpdenote the kernel of the Frobenius endomorphism on the additive group scheme$\mathbb {G}_{a}$Gaover an algebraically closed field of characteristicp. The Verlinde category Verpis thesemisimplificationof the representation category Repαp, and Verpcontains the category of super vector spaces as a full subcategory. Each exceptional Lie superalgebra we construct is realized as the image of an exceptional Lie algebra equipped with a nilpotent derivation of order at mostpunder the semisimplification functor from Repαpto Verp.