Two geometric proofs of the classification of algebraic supergroups with semisimple representation theory

We present two novel proofs of the known classification of connected affine algebraic supergroups $G$ such that $\mathrm{Rep}G$ is semisimple. The proofs are geometrically motivated, although both rely on an algebraic lemma that characterizes $\mathfrak{o}\mathfrak{s}\mathfrak{p}(1\mathrm{\mid }2n)$ amongst simple Lie superalgebras.