Semisimplifying Frank Lie algebras

Abstract

The Frank Lie algebras are simple Lie algebras that only occur over fields of characteristic 3. These come equipped with distinguished inner derivations that make them algebras in the category $\textbf{Rep}(\alpha_3)$. We apply the semisimplification functor to these Frank Lie algebras and obtain modular contact Lie superalgebras. We also obtain a class of simple $J$-ternary algebras whose associated Jordan algebras are not simple.