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

Abstract

Using tensor categories, we present new constructions of several of the exceptional simple Lie superalgebras with integer Cartan matrix in characteristic p = 3 and p = 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, let αp denote the kernel of the Frobenius endomorphism on the additive group scheme Ga\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb {G}_{a}$\\end{document} over an algebraically closed field of characteristic p. The Verlinde category Verp is the semisimplification of the representation category Repαp, and Verp contains 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 most p under the semisimplification functor from Repαp to Verp.