Naturality and innerness for morphisms of compact groups and (restricted) Lie algebras

TL;DR

This paper characterizes extended derivations and endomorphisms of (restricted) Lie algebras and compact groups, showing they are either trivial or inner, thus revealing their naturality and innerness properties.

Contribution

It proves that extended endomorphisms of (restricted) Lie algebras are limited to identity or zero maps, and establishes a canonical bijection between a Lie algebra and its extended derivations.

Findings

Extended endomorphisms of restricted Lie algebras are only identity or zero maps.

Lie algebra in characteristic zero or positive characteristic is in bijection with its extended derivations.

Components of extended endomorphisms of compact connected groups are either trivial or inner automorphisms.

Abstract

An extended derivation (endomorphism) of a (restricted) Lie algebra $L$ is an assignment of a derivation (respectively) of $L'$ for any (restricted) Lie morphism $f:L\to L'$, functorial in $f$ in the obvious sense. We show that (a) the only extended endomorphisms of a restricted Lie algebra are the two obvious ones, assigning either the identity or the zero map of $L'$ to every $f$; and (b) if $L$ is a Lie algebra in characteristic zero or a restricted Lie algebra in positive characteristic, then $L$ is in canonical bijection with its space of extended derivations (so the latter are all, in a sense, inner). These results answer a number of questions of G. Bergman. In a similar vein, we show that the individual components of an extended endomorphism of a compact connected group are either all trivial or all inner automorphisms.