Extensions and automorphisms of Lie algebras

TL;DR

This paper investigates the conditions under which automorphisms of subalgebras and quotient algebras extend to automorphisms of the entire Lie algebra, using cohomology to identify obstructions and providing explicit criteria for certain free nilpotent Lie algebras.

Contribution

It establishes a cohomological obstruction theory for automorphism extension in Lie algebra extensions and derives explicit conditions for free nilpotent Lie algebras.

Findings

Obstruction for automorphism extension lies in Lie algebra cohomology (B;A).

Derived a four-term exact sequence relating automorphisms, derivations, and cohomology.

Provided explicit necessary and sufficient conditions for automorphism extension in specific free nilpotent Lie algebras.

Abstract

Let $0 \to A \to L \to B \to 0$ be a short exact sequence of Lie algebras over a field $F$, where $A$ is abelian. We show that the obstruction for a pair of automorphisms in $\Aut(A) \times \Aut(B)$ to be induced by an automorphism in $\Aut(L)$ lies in the Lie algebra cohomology $\Ha^2(B;A)$. As a consequence, we obtain a four term exact sequence relating automorphisms, derivations and cohomology of Lie algebras. We also obtain a more explicit necessary and sufficient condition for a pair of automorphisms in $\Aut\big(L_{n,2}^{(1)}\big) \times \Aut\big(L_{n,2}^{ab}\big)$ to be induced by an automorphism in $\Aut\big(L_{n,2}\big)$, where $L_{n,2}$ is a free nilpotent Lie algebra of rank $n$ and step $2$.