The Jacobson--Morozov theorem and complete reducibility of Lie subalgebras

TL;DR

This paper extends classical Lie algebra theories to positive characteristic fields, classifies subalgebras based on G-complete reducibility, and clarifies conditions for extending nilpotent elements to sl_2-triples.

Contribution

It advances the understanding of Lie subalgebras in positive characteristic, providing classifications and characterizations that generalize classical theorems.

Findings

Classifies non-G-completely reducible subalgebras for exceptional groups in characteristic p≥5.

Shows G-complete reducibility characterizes bijections between sl_2-subalgebras and nilpotent orbits.

Identifies a unique case in G_2 where a nilpotent element cannot extend to an sl_2-triple when p=3.

Abstract

In this paper we determine the precise extent to which the classical sl_2-theory of complex semisimple finite-dimensional Lie algebras due to Jacobson--Morozov and Kostant can be extended to positive characteristic. This builds on work of Pommerening and improves significantly upon previous attempts due to Springer--Steinberg and Carter/Spaltenstein. Our main advance arises by investigating quite fully the extent to which subalgebras of the Lie algebras of semisimple algebraic groups over algebraically closed fields k are G-completely reducible, a notion essentially due to Serre. For example if G is exceptional and char k=p\geq 5, we classify the triples (\h,\g,p) such that there exists a non-G-completely reducible subalgebra of \g isomorphic to \h. We do this also under the restriction that \h be a p-subalgebra of \g. We find that the notion of subalgebras being G-completely reducible effectively characterises when it is possible to find bijections between the conjugacy classes of sl_2-subalgebras and nilpotent orbits and it is this which allows us to prove our main theorems. For absolute completeness, we also show that there is essentially only one occasion in which a nilpotent element cannot be extended to an sl_2-triple when p\geq 3: this happens for the exceptional orbit in G_2 when p=3.