Maximal subalgebras of the exceptional Lie algebras in low characteristic

Abstract: In this thesis we consider the maximal subalgebras of the exceptional Lie algebras in algebraically closed fields of positive characteristic. This begins with a quick recap of the article by Herpel and Stewart which considered the Cartan type maximal subalgebras in the exceptional Lie algebras for good characteristic, and then the article by Premet considering non-semisimple maximal subalgebras in good characteristic. For $p=5$ we give an example of what appears to be a new maximal subalgebra in the exceptional Lie algebra of type $E_8$. We show that this maximal subalgebra is isomorphic to the $p$-closure of the non-restricted Witt algebra $W(1;2)$. After this, we focus completely on characteristics $p=2$ and $p=3$ giving examples of new non-semisimple maximal subalgebras in the exceptional Lie algebras. We consider the Weisfeiler filtration associated to these maximal subalgebras and leave many open questions. There are one or two examples of simple maximal subalgebras in $F_4$ for $p=3$ and $E_8$ for $p=2$.