On filiform Lie algebras. Geometric and algebraic studies

TL;DR

This paper studies the geometric and algebraic properties of finite-dimensional filiform Lie algebras, focusing on conditions for contact and symplectic structures, and describes their algebraic variety structure in small dimensions.

Contribution

It provides necessary and sufficient conditions for filiform Lie algebras to admit contact or symplectic forms and characterizes the algebraic variety of such algebras in small dimensions.

Findings

Conditions for contact and symplectic structures on filiform Lie algebras

Description of the algebraic variety of filiform Lie algebras in small dimensions

Identification of algebraic components in the variety

Abstract

A finite dimensional filiform K-Lie algebra is a nilpotent Lie algebra g whose nil index is maximal, that is equal to dim g -1. We describe necessary and sufficient conditions for a filiform algebra over an algebraically closed field of characteristic 0 to admit a contact linear form (in odd dimension) or a symplectic structure (in even dimension). If we fix a Vergne's basis, the set of filiform n-dimensional Lie algebras is a closed Zariski subset of an affine space generated by the structure constants associated with this fixed basis. Then this subset is an algebraic variety and we describe in small dimensions the algebraic components.