Prolongation rigidity of sub-free Lie algebras

TL;DR

This paper proves that certain fundamental graded nilpotent Lie algebras are prolongation rigid under specific conditions, with exceptions related to maximal parabolic subalgebras of simple Lie algebras.

Contribution

It establishes a criterion for prolongation rigidity of fundamental graded nilpotent Lie algebras based on the irreducibility of their 0-th Tanaka prolongation.

Findings

Proves prolongation rigidity when $ ext{pr}_+(rak{m})=0$ under given conditions.

Identifies exceptions as negative gradations of maximal parabolic subalgebras.

Provides a classification related to simple Lie algebras.

Abstract

We prove that if the 0-th Tanaka prolongation $\mathfrak{g}_0=\mathfrak{der}_0(\mathfrak{m})$ of a fundamental graded nilpotent Lie algebra $\mathfrak{m}=\mathfrak{g}_{-s}\oplus\dots\oplus\mathfrak{g}_{-1}$ is irreducible on $\mathfrak{g}_{-1}$, then $\mathfrak{m}$ is prolongation rigid: $\text{pr}_+(\mathfrak{m})=0$. The only exceptions are given by negative gradations of maximal parabolic subalgebras of a simple Lie algebra.