The index of nilpotent Lie algebras

TL;DR

This paper investigates the index of nilpotent Lie algebras, providing explicit formulas for various classes such as free-nilpotent, graph, and filiform nilpotent Lie algebras, which are significant in multiple mathematical contexts.

Contribution

It offers new explicit formulas for the index of several classes of nilpotent Lie algebras, advancing understanding in Lie algebra invariants.

Findings

Explicit formulas for free-nilpotent Lie algebras of class two and three

Formulas for graph Lie algebras of solvability class two

Results for filiform nilpotent Lie algebras

Abstract

The index of a Lie algebra is an important invariant which arises in several areas, e.g. in the study of coadjoint orbits for a Lie group, in invariant theory and in representation theory. We study the index for several classes of nilpotent Lie algebras. In particular, we give explicit formulas for free-nilpotent Lie algebras of nilpotency class two and three, or of solvability class two, for graph Lie algebras, and for filiform nilpotent Lie algebras.