Prolongation rigidity of sub-free 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 $\mathfrak{m}={\mathfrak{g}}_{-s}\oplus \cdots \oplus {\mathfrak{g}}_{-1}$ 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.

This version is extended with a lemma, a corollary and a remark, providing more details. A minor error in an example is corrected and a reference is added