The Nash-Moser Theorem of Hamilton and rigidity of finite dimensional nilpotent Lie algebras

TL;DR

This paper applies the Nash-Moser theorem to study deformations and rigidity of finite-dimensional nilpotent Lie algebras, providing new insights into their structure and identifying rigid cases in low dimensions.

Contribution

It introduces a new framework using the Nash-Moser theorem for analyzing deformations of nilpotent Lie algebras and identifies rigid algebras and curves in low-dimensional cases.

Findings

Identification of rigid Lie algebras in dimensions up to 7.

Construction of rigid curves in 3-step nilpotent Lie algebras of dimension 7.

Clarification and correction of previous results in the literature.

Abstract

We apply the Nash-Moser theorem for exact sequences of R. Hamilton to the context of deformations of Lie algebras and we discuss some aspects of the scope of this theorem in connection with the polynomial ideal associated to the variety of nilpotent Lie algebras. This allows us to introduce the space $H_{k-nil}^2(\mathfrak{g},\mathfrak{g})$, and certain subspaces of it, that provide fine information about the deformations of $\mathfrak{g}$ in the variety of $k$-step nilpotent Lie algebras. Then we focus on degenerations and rigidity in the variety of $k$-step nilpotent Lie algebras of dimension $n$ with $n\le7$ and, in particular, we obtain rigid Lie algebras and rigid curves in the variety of 3-step nilpotent Lie algebras of dimension 7. We also recover some known results and point out a possible error in a published article related to this subject.