A new invariant of quadratic Lie algebras

TL;DR

This paper introduces a novel invariant for quadratic Lie algebras, providing a comprehensive classification of singular cases linked to orthogonal group orbits, advancing understanding in Lie algebra theory.

Contribution

It defines a new invariant for quadratic Lie algebras and classifies singular cases through orbit analysis, offering new insights into their structure.

Findings

Complete classification of singular quadratic Lie algebras

Connection between invariants and O(n)-adjoint orbits

New invariant distinguishes non-vanishing cases

Abstract

We define a new invariant of quadratic Lie algebras and give a complete study and classification of singular quadratic Lie algebras, i.e. those for which the invariant does not vanish. The classification is related to $\mathrm{O}(n)$-adjoint orbits in $\mathfrak{o}(n)$.